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and 7~(t») = (7o^|n-)|s- The key point is to show that these trace operators are independent on the choice of fi+ or fi~; This can be proved at least for a polygonal boundary of ft, i.e. if (H) holds; see [ANG 99]. D Lemma 2 (Jump of traces on S in Hl(Q)) If (H] holds, then : PROOF. We get for all v G Hl(Q), using the continuity of traces in Lemma 1 We now define the Hilbert space W = {v € L 2 (ft) n Hl(ty; 7(7;) = u|f = 0} equipped with the usual inner product and associated norm in H l ( f y . Then we prove the following proposition by considering the variational formulation of the problem (1-3) in ft as below : Find u G W such that Vv G W, with the convective part Tc — u
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219
Theorem 1 (Existence and uniqueness for the crack model) // the assumptions (H), (Al), (A2) and (A3) hold, the problem (1-3) with g 6 £ 2 (£) and f £ L 2 (Q) has a unique weak solution u £ W satisfying (4) Vu £ W, such that:
PROOF. The bilinear symmetric form in the left-hand side of (4) is clearly coercive in W. By using Lemma 2, we easily show the continuity of both this bilinear form in W x W and the linear form, i.e. the right-hand side of (4), in W. Then we apply the Lax-Milgram theorem which completes the proof. D 3. Finite volume approximation on general meshes For the sake of simplicity, we only consider here the case of an isotropic tensor of diffusion, i.e. a scalar coefficient variable in space a(x) satisfying (Al). The general case of a diffusion matrix a(x) requires an additional condition of quasi-regularity of the mesh, e.g. [EYM 97]. Definition 1 (Admissible meshes) Let 17 = 17 U £ be an open bounded polygonal subset of IRd, d — 2, or 3 satisfying (H) in Lemma 1. An admissible finite volume mesh of 17; denoted by T, is given by a family of "control volumes", which are open polygonal convex subsets of 17 , a family of subsets of 17 contained in hyperplanes of JRd, denoted by 8 (these are the edges (twodimensional) or sides (three-dimensional) of the control volumes), with strictly positive (d — I)-dimensional measure, and a family of points of 17 denoted by P satisfying the following properties (in fact, we shall denote, somewhat incorrectly, by T the family of control volumes): (i) The closure of the union of all the control volumes is 17. (ii) For any K £ T, there exists a subset £K of 8 such that OK = K \ K = U0. e< r K cr. Furthermore, £ = UK^T^-K(Hi) For any (K, L) £ T2 with K ^ L, either the (d— I)-dimensional Lebesgue measure of K Pi L is 0 or K fl L — W for some a £ £, which will then be denoted by K\L. (iv) The family P = (XK)K^T is such that XK £ K (for all K £ T) and, if a ~ K\L, it is assumed that XK ^ XL, and that the straight line T*K,L going through XK and XL is orthogonal to K\L. (v) There exists £^ C. £ such that E = U ff ££ s <7. (vi) For any a £ £ such that a C 5H = d$l U E, let K be the control volume such that a G £K- If XK £ ai ^ ^K,a be the straight line going through XK and orthogonal to a, then the condition T>K,a H a ^ 0 is assumed; let y
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Finite volumes for complex applications
In the sequel, the following notations are used. The mesh size is defined by: hj- = sup{diam(K), K € T}. For any K 6 T and a 6 £, m(K) is the ddimensional Lebesgue measure of K (it is the area of K in the two-dimensional case and the volume in the three-dimensional case) and m(cr) the (d — \)dimensional measure of a. The set of interior (resp. boundary) edges is denoted by £int (resp. £exij, that is £int — [a G £; a $_ dfr} (resp. £ext = {cr e £; o C d$i}). The set of neighbours of K is denoted by N(K], that is J\f(K) = [L £ T; 3cr G £K, o" — K r\ L}. If o~ = K\L, we denote by da or d^.\L the Euclidean distance between XK and XL (which is positive) and by G?K,
and *< = ^K) IKb(x)dx' fK = ^K) fKf(x">dx' P* = ^7) L P(X"> ds(x^ 9* ~ ^y f a 9 ( x ) d s ( x ) , or simply by aK = a(xK], bK = b(xK), fK = /(XK}, Pa = p(y<j} or ga = g(ya} if the coefficients are regular enough to define these quantities. Define X(T} as the set of functions from fi to IR which are constant over each control volume of the mesh. For v € X(T] such that VK denotes the value taken by v on any control volume K € T, define the discrete HQ(&) norm by :
where
and
3.1. Finite volume method with fracture resistances As in [ANG 89, ANG 98] for a rectangular mesh, the scheme is derived with respect to p|s instead of (3\% because it is more practical since the case when p\Y, = 0 and g\
For a £ £ e xt> the Dirichlet boundary condition is written. Finally, the numerical scheme reads in the following synthetic form: for all K £ T,
3.2. Convergence analysis and error estimate Theorem 2 (Error estimate of the FV scheme) // ("H) and the assumptions (Al), (-42), (.A3) hold, let u £ W be the unique weak solution of the problem (1-3) with g £ L 2 (£), / £ L 2 (f)). Let T be an admissible mesh for the discretization of that problem, in the sense of Definition 1. Then, there exists a unique discrete solution (UK)K^T to the finite volume scheme (6-7-8-9-10-11). Moreover, assume piecewise sufficiently regular data: \/K £ T', f\K £ C Q ( K ] , b\K € Cl(K) and O\K in Cl(K], and Vcr £ £•%, P\a o,nd g\a in Cl(a], and such that the solution u £ Hl($l) satisfies U\K £ C2(K], for any K £ T. Define the error ej- £ X(T) by e-r(x] = U(XK) — UK for a.e. x £ K, and for all K £ T. Then, there exists C > 0 only depending on u, a, b, p and SI such that the following error estimates hold for the discrete HQ and L2 norms: and SKETCH OF PROOF. It is inspired from [EYM 97]. With the conservativity property, a discrete coercivity lemma is proved which ensures both the existence and uniqueness of the discrete solution via a discrete maximum principle and the stability in the sense of finite differences. By using Taylor expansions and the regularity assumptions, the weak consistency in O(h-j-) of the approximate volumic terms and the numerical fluxes, specially for (3) on S, is proved. Then it yields the error estimate for the discrete HQ norm. Finally, the L2 estimate follows from a discrete Poincare inequality. D Remark 1 (Convergence and weaker assumptions) With no more regularity on the solution than u £ Hl(Q), it is possible to prove directly the convergence of the scheme by using some compactness results and a discrete trace
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lemma established in [EYM 97]. Besides, error estimates in O(hj-) like above, also hold for weaker regularity assumptions on the solution, i.e. u G Hl(£l] and U\K £ H^(K) for any K £ T. It requires the use of Taylor expansions with integral residuals to prove the weak consistency. See [ANG 99] for the details. References [ANG 89] [ANG 89b]
[ANG 97]
[ANG 98]
[ANG 98b]
[ANG 99] [COU 96]
[EYM 97]
[HEI 87] [HER 95]
[HER 96]
[LAZ 96]
[SAM 78] [SHA 96]
ANGOT PH., Ph.D. Thesis, University of Bordeaux I, april 1989. ANGOT PH., Modeling and visualization of thermal fields inside electronic systems under operating conditions, IBM Technical Report, TR-47095. 70 p., april 1989. ANGOT PH., Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows, M2 AS Math. Meth. in the Appl. Sci., Preprint 1997 (to appear). ANGOT PH., Finite volume methods for non smooth solution of diffusion models; Application to imperfect contact problems, in Advances in Numerical Methods and Applications, O.P. Iliev et al. (Eds), Proceedings 4th Int. Conf. NMA '98, Sofia, World Scientific Pub., 1999. ANGOT PH., Mathematical and numerical modelling for a fictitious domain method with penalized immersed boundary conditions, Preprint HDR Thesis - Univ. Mediterranee, 1998 (submitted). ANGOT PH., GALLOUET TH., HEREIN R., in preparation. COUDIERE Y., VILA J.-P., VILLEDIEU P., Convergence of a finite volume scheme for a diffusion problem, in Finite Volumes for Complex Applications - Problems and Perspectives, F. Benkhaldoun and R. Vilsmeier (Eds), Hermes, p. 161-168, 1996. EYMARD R., GALLOUET TH., HEREIN R., Finite Volume Methods, in "Handbook of Numerical Analysis", P.G. Ciarlet and J.L. Lions (Eds), North-Holland, Preprint 1997 (to appear). HEINRICH B., Finite Difference Methods on Irregular Networks, Int. Series Numer. Math. 82, Birkhauser Verlag, Basel, 1987. HEREIN R., An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh, Num. Meth. for P.D.E., 11:165-173, 1995. HEREIN R., Finite volume methods for diffusion convection equations on general meshes, in Finite Volumes for Complex Applications - Problems and Perspectives, F. Benkhaldoun and R. Vilsmeier (Eds), Hermes, p. 153-160, 1996. LAZAROV R.D., MISHEV I.D., Finite volume methods for reactiondiffusion problems in Finite Volumes for Complex Applications Problems and Perspectives, F. Benkhaldoun and R. Vilsmeier (Eds), Hermes, p. 233-240, 1996. SAMARSKII A.A., ANDREEV V.B., Methodes aux Differences pour Equations Elliptiques, Editions MIR Moscou, 1978. SHASHKOV M., Conservative finite-difference methods on general grids, CRC Press New York, 1996.
Application and analysis of finite volume upwind stabilizations for the steady-state incompressible Navier-Stokes equations
Lutz Angermann Institut fiir Analysis und Numerik Fakultdt fur Mathematik Otto-von-Guericke-Universitdt Magdeburg PF 4120 D-39016 Magdeburg, Germany
ABSTRACT Among the various difficulties in the numerical solution of NavierStokes equations, the case of high Reynolds numbers requires extra care in the choice of a concrete discretization scheme. The specific treatment of the nonlinear connective term by means of finite-volume-based approaches within the framework of conforming or nonconforming finite element m.ethods forms one class of practically relevant stabilization techniques. The aim of the contribution is to describe a general approach to the design of those methods in multidimensional situations and to extract the basic principles of their analysis. Key Words: finite volume upwind method, incompressible Navier-Stokes equations
1. Introduction
Let c > 0 be a real number, Q C ffi d with d = 2 o r c / = 3 b e a bounded, polyhedral domain with Lipschitz continuous boundary and / : Q —>• Md be a given vector field. With V := W$(tt)d and Q := L*(tt) := {q € L2(Q) : (q, 1) = 0}, the following weak Navier-Stokes problem is considered: Find (ti, p) = (it 1 ,... , ud, p) € V x Q such that it holds:
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Finite volumes for complex applications
where the trilinear form n : V3 —)• M is defined as n(w,u,v) := ((w • V)it,u) and (•, •) denotes the inner product of L^Q] or L2(Q)d• Now let Vh,Qh be members of two families of finite element spaces approximating, in certain sense, the elements of V, Q. Here, the discrete spaces need not be subspaces of V, Q. While it is not difficult to replace the forms (Vit, Vv) and (p, V • v) by their "broken" counterparts on the underlying partitions 7/i of Q
the trilinear form n has to be defined in a more careful way. In the book [RST 96, Sect. IV.2], the authors pointed out that the following three properties of the discrete form n/j : Vft3 —>• M are sufficient conditions to establish convergence of the numerical method: Semidefiniteness: Lipschitz-continuity: Consistency:
where || • ||/j is a norm on Vh and //, : W$(Q}d fl V —>• VA denotes some interpolation operator. The following general approach for the design of the discrete trilinear form satisfying the above conditions is based on three different partitions of Q. One of the partitions is denoted by Th and is either a triangulation (i.e. it consists of d-simplices) or a block-partition (i.e. it consists of convex quadrilaterals (d — 2) or convex hexahedra (d — 3)). It is assumed that Th is admissible in the usual sense, i.e. two elements of the partition are allowed to have in common either a vertex or an edge or, if d — 3, a face. Using the notation T for the elements of Th, the parameter h of the partition is defined as usual: If T has the diameter HT, then h := max h?- Notice that this partition is related T&TH to the approximation Uh of the unkonwn u; in certain situations the partition for the discrete unknown ph may differ from ThNext, on each partition Th a (not necessarily conforming) finite element space is defined, the elements of which are piecewise polynomially of maximal degree / (E N U J O } . In particular, the polynomial space on T may be incomplete, especially for quadrilateral/hexahedral elements. Given some finite element space, the corresponding set of functionals (global degrees of freedom) naturally splits into Langrangian functionals and others, where a Langrangian functional is defined via point values of its argument. Therefore, considering these Langrangian functionals, a collection of (global) nodes can be associated in a natural way. This collection of nodes can be subdivided into the class of nodes lying on element boundaries and the class of nodes belonging to the interior of some element. For example, the second
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225
class is empty for all serendipity elements, but for the mini element it contains all element barycenters. Let A.g denote the set of indices of all the nodes from the first class. The subset A.g C A.g contains, by definition, the indices of interior (w.r.t. Q) nodes only. Finally, declare d\g := A f f \ A g and let \gT C As contain the (global) indices of the nodes belonging to the element T. Due to the boundary conditions, the above finite element space is restricted to elements satisfying a discrete boundary condition, i.e. we set
The distance between two nodes #,, Xj ( i , j £ A9) is denoted by dij. Now, an important observation is that the index set A.g can be decomposed into two disjoint subsets A, A such that A9 — A U A and A fl A =0. Such a decomposition can be generated, for example, by a hierarchical decomposition of the finite element space Shi into a "lower degree part" and its linear complement. Then A corresponds to the nodes of the first part and A to the complement's nodes. Obviously, this decomposition induces similar ones of A5, d\.g and A^, respectively. If Shi is a space of elements of low degree, then it is allowed that the decomposition is trivial, i.e. the complement may be the trivial space consisting of the zero element only. In this case, A is empty by definition. As an example, in the case when Shi is built by means of conforming /Velements (Taylor-Hood elements, d = 2), we have the decomposition
where Shi is the space of conforming Pi-elements (Courant elements) and S^ is the space of so-called "bump functions", i.e. the set of the piecewise quadratic nodal basis functions associated with the edge midpoints. To describe the discretization of the trilinear form n, a further partition of Q is needed. This second partition 7^* consists of subdomains Q2 C ^, the boundary part Q D d&i of which is the union of subsets of (d — l)-dimensional hyperplanes. An incidence relation between these two partition is defined by the help of the nodes of Th, i.e. each f22 should correspond to one node Xi and vice versa. Further, for all indices i, _;' G A, j ^ z, let F,j := Q, D fij and mzj := measrf_i ( F z j ) . Then it makes sense to introduce the index set
As a consequence of these definitions, the following representation of dQ,i is valid: d£li — M F,-j, i 6 A. Moreover, there are the obvious symmejeA,
try relations djj = dij, Fj,- = F,-j, ra^- = m,-j. The boundary parts F;J can be structured in a finer way: F^- := F,-j fl T for T € Th- Analogously, mj := meas d _i (F^) . Now we formulate a set of geometrical assumptions.
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(Al) There exists a constant C > 0 such that, for all /, / £ A, it holds (A2) There exists a third partition {&?•}. . -r-. ,. „ T >0n of fi such that the l J J z , j € A,t7=j, T:m^ subdomains Q^ have the following properties:
(iii) Each ^ij- can be decomposed into a finite number ifj r of pairwise ™ i
T
i
f i
disjoint open c/-simplices Q-' such that Q 2 = U/=i ^W an-d, for any / £ [1,/^-JTv? ^,-/ is the image of a fixed (reference) simplex T under a regular affine transformation, where the pre-image F of F. ' := FJJnf2,-j' does not depend on the particular values of z, j, T, /. rp
,
T1^
(iv) On each F - • ' , the unit outer (w.r.t. Q z -) normal i/-•' is constant. T1
I
T~>
/
The diameter of the simplex 0,- •' is denoted by /i-•' . Furthermore, we set m-'T I := measd_i I( F -T- 'l j\ I. Finally, we define 17jj := int I (A3) There exists a constant C > 0 such that, for all z,j £ A, it holds
(A4) There exists a constant C > 0 such that, for all i, j £ A, T £ 7/j, / £ [I,/^-]AT, it holds mfy c?,-j < Cmeas^ fSl f y J . (A5) There exists a constant C > 0 such that, for all i,j £ A, T £ 7/i, / £ [1, /5]jv, it holds (/ij'')4 < Cmeasd (^'') . A dimensional analysis of the quantities appearing in the Assumptions (Al), (A3), (A5) easily shows that these conditions are not very restrictive.
2. Discretization (Treatment of the trilinear form n)
Because of
the description of the discretizaV
tion can be restricted to the scalar case. So let w £ Wl(£l}d be such that
Numerical analysis 227
V • w — 0 and define, for w, v £ W / 2 1 (fi), the form
It is not difficult to see that ns can be represented equivalently as
Taking into consideration the condition V • w •= 0, it is not so far to omit the last term, i.e. we get
where This is the starting point for the discretization. Suppose there is some control function r : M —> [0, 1] satisfying the following properties: • r(z) is isotone for all z,
zr(z) is Lipschitz-continuous on the whole real axis. /*
Furthermore, set 7,-; := m"31 / J
JrtJ
/ **V ' " ft ' ' \
v • w ds and ra := r(
\
IJ IJ
s
}. We mention
J
that 7jj is antisymmetric, i.e. 7^,- = — 7^-. Moreover, in the definition of 7;j, the value of Jr v • w ds can be replaced by certain approximation which has to satisfy, among natural error estimates, the above antisymmetry condition. Then, by standard arguments in the derivation of finite volume discretizations (cf. [ANG 95]), we can write
Thus we get
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Finite volumes for complex applications the quantities
Now, redefining for as well as
we set
Returning to the original form n, we set for
COROLLARY 1 // the control function r satisfies (P5), then it holds
Typical examples of the control functions are
(full upwind
(exponentially fitted scheme),
scheme), (Samarskij's scheme).
Finally, some discrete forms and operators have to be introduced. For
we set where
The extension of these definitions to the case of Rd-valued functions is obvious and will not be denoted separately. By Ih '• W%(£1] —>• Shi, an interpolation operator is denoted, whereas Lh '• C(fi) —>• ico(^) stands for a so-called lumping procedure. That is, the image of Lh is the subspace consisting of functions being constant on the elements of the secondary partition 7^*. Concrete properties of these operators are collected in the subsequent assumptions.
3. Properties of the discrete forms In order to verify the required properties of Lipschitz-continuity and consistency of 77,5/2, we formulate further assumptions.
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(A6) There exists a constant C > 0 independent of h such that, for all v/, G SM, it holds
(A7) For arbitrary p G [1,6], there exists a constant C > 0 such that, for all Vh € SMJ*> holds (A8) There exists a constant C > 0 such that, for arbitrary p G [1,6] and for all Vh G SM, it holds
Since A is non-empty, in general, ||^/jV/,||o,p,n is on ly a seminorm on Shi(A9) There exists a constant C > 0 such that for all v/, G Shi, it holds
(A10) There exists a constant C > 0 such that (i) for all v G W%(Q), it holds
(ii) for all v G W%(ti) and all T G T/,, it holds
(iii) for arbitrary p G (rf, 6], u G W 1 ^)
and all T G TA, it holds
(All) There exists a constant C > 0 such that (i) for arbitrary p G (d, 6], v G W^(n) and all T G TA, it holds:
(ii) for all v G W$(tt) and all T G Th, it holds
(A 12) There exists a constant C > 0 such that, for all v G W$(£l) and all T G T/M it holds
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Finite volumes for complex applications
Now, the folllowing results can be proved (for details, see [ANG 98]). LEMMA 1 Suppose (Al), (A2), (A3), (A4), (AS), (A6), (A7), (A8). Then, for arbitrary Wh^Zh 6 Vh and u/^v/i G Shi the estimate
holds, where C > 0 is a constant which does not depend on h. LEMMA 2 Suppose (A2), (A4), (A6), (Al), (A8), (A9), (AW), (All), (A 12). Then, for any w £ Vl / 2 2 (^) rf fl V satisfying V • w = 0, any u £ 0
W-2(£l] n W£(£l} and any element Vh € Shi the estimate
holds, where C > 0 is a constant which does not depend on h.
4. Application The above approach can be used to give an alternative proof of the convergence properties of Schieweck's family of nonconformingquadrilateral/hexahedral elements [SCH 97] which find successful application in parallel Navier-Stokes codes. Details for the case of the so-called Pi-parametric element are described in [ANG 98]. 5. Bibliography [ANG 95]
ANGERMANN, L. Error estimates for the finite-element solution of an elliptic singularly perturbed problem. IMA J. Numer. Anal, 15, 1995, p.161-196.
[ANG 98]
ANGERMANN, L. Error analysis of upwind-discretizations for the steady-state incompressible Navier-Stokes equations. Fakultat fur Mathematik, Otto-von-Guericke-Universitat Magdeburg, Preprint Nr. 33, 1998.
[RST 96]
Roos, H.-G., STYNES, M. AND TOBISKA, L. Numerical methods for singularly perturbed differential equations. Springer- Verlag, Berlin-Heidelberg-New York, 1996.
[SCH 97]
SCHIEWECK, F. Parallele Losung der stationaren inkompressiblen Navier-Stokes Gleichungen. Habilitationsschrift, Fakultat fur Mathematik, Otto-von-Guericke-Universitat Magdeburg, 1997.
A new cement to glue non-conforming grids with Robin interface conditions
Yves Achdou Insa Rennes, 20 Av. des Buttes de Coesmes 35043 Rennes, France
Caroline Japhet, Frederic Nataf CMAP, Ecole Polytechnique 91128 Palaiseau, France
Yvon Maday Laboratoire d'Analyse Numerique, Universite Pierre et Marie Curie 4, place Jussieu, 75252 Paris Cedex 05, France
ABSTRACT We propose and analyse a domain decomposition method based on Schwarz type algorithms that allows for an extension to optimized interface conditions on nonconforming grids. We consider the convection-diffusion equation discretized by a finite volume method. The nonconforming domain decomposition method is proved to be well-posed and the error analysis is performed. Then we present numerical results that illustrate the method. Key Words: Domain decomposition methods, optimized artificial interface conditions, non-conforming grids, convection-diffusion problems, finite volume methods, parallel computing, High Performance Computing.
1. Introduction The goal of our project is to design domain decomposition methods based on the use of optimized interface conditions on non-matching grids. The original Schwarz algorithm is based on a decomposition of the domain 17 into overlapping subdomains and the solving of Dirichlet boundary value problems in the subdomains. It has been proposed in [L 89] to use more general boundary conditions for the subproblems in order to use a non-overlapping decomposition of
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Finite volumes for complex applications
the domain. The use of exact artificial boundary conditions as interface conditions leads to an optimal number of iterations, see [HTJ 88], [NRdeS 95]. As these boundary conditions are pseudo-differential, "low frequency" approximations of these conditions have then be proposed, see [D 93], [NR 95], [GGQ 96]. In [J 96] approximations which minimize the convergence rate of the algorithm are proposed, and increase dramatically the convergence speed of the method. The mortar method, first introduced in [BMP 89], which enables the use of non-conforming grids, can't be used easily with optimized interface conditions in the framework of Schwarz type methods. The goal of our work is to design and study a non-conforming domain decomposition method which allows for the use of Robin interface conditions (J^ +a) for Schwarz type methods. We consider the convection-diffusion equation
discretized by a finite volume method where rj and v are positive but arbitrary small and a is the vector field. We first consider the symmetric definite positive case (a = 0) and then in § 5 the convective case (a ^ 0). 2. Domain Decomposition at the continuous level
Let fi be a bounded domain in Rd for d > I and 77 > 0. We consider the following problem: Find u such that
The domain fi is decomposed into N non-overlapping subdomains, fi = ^Ji
An iterative method for solving the above domain decomposition method is:
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233
The well-posedness and convergence of the above problems and algorithm have been studied in [L 89]. We are interested in the discretization of (4) by a finite volume scheme with non matching grids on the subdomains's interface. 3. Finite volume discretization
The scheme is taken from [H 95]. On each domain 0; let Ti be a set of closed polygonal subsets of 0; such that f^ = \JxeTiK and £^ii the set of edges associated with Ti, i.e. a set of closed subsets of dimension d such that for any (K, K1} G T? with K ^ K1, one has either K n K' = 0, dim(K n K'} < d - 1 and K n K' G £ n .. In this case, dK n dK1 is denoted by [K,K']. We also assume that no edge intersects both dtli/dQ, and <9fi; n 80,. We shall use the following notations: Let 6i be an edge of £^ii located on the boundary of fii, K(ti) denotes the control cell K G Ti such that e; G K. £iD is the set of edges such that dtl n dtl{ = U ee £. D e. Let us recall that a Dirichlet boundary condition is imposed on this part of the boundary. £j is the set of edges such that d£li/d$l — Ueg^e. Let us recall that a Robin interface condition is imposed on this part of the boundary. £(K] denotes the set of the edges of K G Ti. £io(K] = £(K] n £iD is the set of the edges of K € Ti which are on <9Q n <9fV £i(K) = £(K}r\£l is the set of the edges of K 6 Ti which are on dtoi/dft. Afl(K) is the set of the control cells adjacent to K: Afi(K) = {K1 G Ti/ Kr\K' e £ n .}. We make the following Assumption 3.1 We assume that there exist points (y e ) ee £ n . on the edges (ye G €.) and points (xK)K&Ti inside the control cells such that for any adjacent control cells, K and K', the straight line [XK,XK>\ is perpendicular to the edge [K, K'] and [XK,XK'] H [K, K'} = {y[K,K']}> o,nd for any edge e G £i U £iD, the straight line [x/^( e ),y € ] is perpendicular to e. It is then possible to write a finite volume scheme for the equation (4). We shall use the primary unknowns (UK}K^T which aim at being approximations to U(XK}- The scheme is obtained by integrating (4) over each control volume K: This relation is discretized by
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Finite volumes for complex applications
where meas([K, K1}} is the measure of [K, K'] and pt is a discretization (defined below) of the normal derivative du/driK on the edge [ K , K ' ] . For an edge [K, K'] common to two control volumes K and K1',
We have then the useful property that PK,K' = —PK',K- For an edge e on the boundary <90, the homogeneous Dirichlet boundary condition (5) is taken into account by
When there is no domain decomposition, this scheme has been analyzed in [H 95] in the more general case of discontinuous coefficients and is proved to be of order one for a discrete Hl-norm. 3.1 Discretization of the interface conditions On each interface edge e € E; of a control volume K — K(e) , we introduce pf and ut related by the relation
Then, the interface condition (6) on an interface edge e G £j of a control volume K — K(e) is discretized by
It will be useful in the error analysis to interpret (15) as a L2 projection. Indeed, let PQ(d£li/d£l] be the set of functions from dSli/d£l into E which are piecewise constant on the interface edges. To any discrete values (t' e ) ee £ { , we associate Ki((vf)e€£i) £ P°(dtti/d£l) its natural piecewise constant extrapolation:
The L2 projection on P°(dQi/d£l} is denoted by P{. With these notations, (15) is equivalent to
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235
For simplicity, (18) will be (sometimes) denoted by abuse of notation
Theorem 3.2 The finite volume discretization defined by (11)-(12)-(13) -(14)(15) is well-posed. 4. Error analysis
For the error analysis, we need the following additional assumption on the interface edges (this assumption is relaxed in [AJNM 99]): Assumption 4.1 a) For any i and any e G £i, ye is the barycenter of e. b) For any i,j, <9Q;ndfij = ^ f £ £ { , t c d t i j t , i-e. d^liCid^tj can be written as the union of edges of £j and of Sj. Let ulK = U(XK) for any control cell K in T(Hj) and for any e € £j£>, let u\ = u(yf) = 0 and p\ — - j ^ ( y f ) . For any interface edge e £ £j, let p\ (resp. u\] be the mean value of -j^- (resp. u). With the notations of § 3.1, this can be rewritten as T i l ( ( p \ } t £ £ i ] = Pi(-j^) and ^((u*)^^) = Pl(u). Let FK be an approximation of order \K\O(h] of the integral of / on the cells K. The solution of the finite volume discretization with FK as a right hand-side is denoted by ulK for any control cell K in T(fJi) and p\ on any edge e G 8i U £io and u\ for any edge e 6 Si. We shall estimate the discrete errors elK — ulK — ulK, e\ — u\ — u\ and q\ = p\ — p\. As for Q, we take a = /3/h^ with 7 > 0. Theorem 4.2 We assume that the solution u of (2) is in C r 2 (A). Let us consider a family of admissible meshes T^
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Finite volumes for complex applications
5. Advection-diffusion problems We consider now the advection diffusion problem with a continuous velocity field a and a viscosity v :
Theorem 5.1 Under the same assumptions as in theorem 4-%> that a 6 C(ft) satisfies V • a = 0, we have the error estimate
ana
provided
6. Numerical results In order to illustrate the use of Robin interface conditions on non matching grids, 2-D test problems were performed with the finite volume scheme analyzed above. We solve the following problem:
with r] and v positive constants. The stopping criterion of the algorithm is that the max norm of the jump of the Robin interface condition is smaller than 10~8. The numerical solution is compared to an exact solution. We choose u(x,y) — x3 y2 + sin(xy). The right-hand side is obtained by applying the operator to u and using an approximate integration formula, the Gauss quadrature rule with four by four nodes per control volume. Thus the error due to inexact integration will be small with respect to the scheme error. A decomposition into 2x2 subdomains is considered. The L^ and discrete H^ (Theorem 5.1) norms are given for successively refined grids: • Case 1: TJ = 1, a = (0,0), v = 1. Initial grid: 9 x 9-8 x 8-7 x 7-6 x 6. • Case 2: r, = 1, a = (y, -x), v = le-2.
Initial grid: 9 x 9 - 8 x 8 - 7 x 7 - 6 x 6 .
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For case 1 (diffusive case), the first order in the discrete H^ norm as expected from the theory seems to be attained only asymptotically for rather fine meshes. The error reduction between the finest two meshes is 1.88. The error reduction factor in the L^ norm is also improved with mesh refinement and seems better than O(h). On the contrary in case 2 (convective case), the error reduction factor deteriorates as the mesh is refined and seems to converge to the expected value \/2-
1/h
II
i ik
Hoc
H^ error reduction
8
0.0269608
0.0626146
16
0.0128424
0.0405976
1.54
32
0.0062293
0.0260017
1.56
64
0.00270477
0.0154682
1.68
128
0.00103779
0.00823524
1.88
Case 1
Table 1: Error vs. mesh refinement - No convection
1/h
II
Hoc
II
k
H\ error reduction
8
0.330713
0.142505
16
0.170031
0.0677237
2.1
32
0.0744696
0.0347217
1.95
64
0.0301526
0.0199572
1.74
128
0.0169002
0.0118619
1.68
Case 2
Table 2: Error vs. mesh refinement - Convection
7. Bibliography [AJNM 99]
ACHDOU Y., JAPHET C., NATAF F., MADAY Y., A new cement to glue non-conforming grids with Robin interface conditions, RI N° 419, CMAP Ecole Polytechnique, may 1999.
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Finite volumes for complex applications
[BMP 89]
BERNARDI C., MADAY Y. AND PATERA A., A new nonconforming approach to domain decomposition: the mortar element method, Nonlinear Partial Differential Equations and their Applications, eds H. Brezis and J.L. Lions (Pitman, 1989).
[D 93]
DESPRES B. Domain decomposition method and the Helmholtz problem. II, Kleinman Ralph (eds) et al., Mathematical and numerical aspects of wave propagation. Proceedings of the 2nd international conference held in Newark, DE, USA, June 7-10, 1993. Philadelphia, PA: SIAM, 197-206 (1993).
[HTJ 88]
HAGSTROM T., TEWARSON R.P. AND JAZCILEVICH A., Numerical Experiments on a Domain Decomposition Algorithm for Nonlinear Elliptic Boundary Value Problems, Appl. Math. Lett., 1, No 3 (1988), 299-302.
[H 95]
HERBIN R., An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh, Numer. Methods Partial Differential Equations 11 , no. 2, 165-173, (1995).
[J 96]
JAPHET C., Optimized Krylov-Ventcell Method. Application to Convection-Diffusion Problems, DD9 Proceedings, John Wiley & Sons Ltd (1996).
[L 89]
LIONS P.L., On the Schwarz Alternating Method III: A Variant for Nonoverlapping Subdomains, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM (1989), 202-223.
[NR 95]
NATAF F. AND ROGIER F., Factorization of the ConvectionDiffusion Operator and the Schwarz Algorithm, M 3 AS, 5, n1, 67-93 (1995).
[NRdeS 95]
NATAF F., ROGIER F. AND DE STURLER E., Domain Decomposition Methods for Fluid Dynamics, Navier-Stokes Equations and Related Nonlinear Analysis, Edited by A. Sequeira, Plenum Press Corporation, pp367-376 (1995).
[GGQ 96]
GASTALDI F., GASTALDI L. AND QUARTERONI A., Adaptative Domain Decomposition Methods for Advection dominated Equations, East-West J. Numer. Math 4, 3, p 165-206 (1996).
Finite Volume Box Schemes
Jean-Pierre CROISILLE Departement de Mathematiques Universite de Metz F-57045 Metz Cedex 01 croisil@poncelet. univ-metz.fr
ABSTRACT: We present the numerical analysis on the Poisson problem of a mixed Petrov-Galerkin Finite Volume scheme for equations in divergence form div(/?(u, Vu) = f, which has been introduced in [CoC 98]. As the original box scheme of Keller, this scheme uses face centered degrees of freedom for the primal unknown u and for the flux (p. The underlying Finite Element spaces are the non-conforming space of Crouzeix-Raviart for the primal unknown and the div-conforming space of Raviart-Thomas for the flux. Optimal order error estimates are derived for the Poisson problem. Key Words: Finite- Volume method, Box Scheme, Box Method, Mixed Method.
1. Introduction The name of "box-scheme" is a generic denomination for several numerical schemes of different origins. It has been introduced primitively by H.B. Keller in the '70 on the 1-D heat equation, [Ke 71]. Generally speaking, the discrete equations are defined in a box-scheme from some kind of averages of the continuous equations on "boxes". Therefore, they are conservative schemes, i.e. schemes which guarantee, for equations in divergence form, an exact conservation of the flux at the level of the box. At least two variants of box-schemes are known in the litterature. The first one has been introduced in the '80 in compressible Computational Fluid Dynamics. As in Keller's scheme, the basic idea is that locating the degrees of freedom at the center of the faces instead at the center of the cells, could be more interesting for an accurate evaluation of
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Finite volumes for complex applications
conservative fluxes, [CDH 83], [CM 86], [Co 92]. Note that this kind of schemes has received much less attention in the CFD Finite Volume communauty than the cell-centered Finite Volume schemes. The second kind of box-schemes is known under the name of "box-method" or "finite volume element method". For the Laplace equation — Au = /, it consists of an approximation of u in a finite-element Pl or Q1-space. The discrete equations are defined by averaging the equation on a dual box surrounding each vertex of the mesh, [BR 87], [Ha 89], [CMM 91], [TS 93]. In this kind of scheme, two meshes are used. The primal one as support of the trial functions, and the dual one for designing the boxes for the discrete conservative equations. This design is in fact similar to the one of the cell-vertex Finite Volume method in CFD, [FS 89]. Concerning the numerical analysis of Finite Volume cell-centered methods, beside the exhaustive direct analysis of [EGH 97], there is by now an attempt in the FEM communauty for interpreting mixed Finite Elements methods as Finite Volume methods, [BMO 96], [Du 97], [YMAC 99]. The scheme presented here can be more or less attached to this kind of study. 2. The Finite Volume Box Scheme In [CoC 98], we introduced a new kind of box-schemes for equations in divergence form. As in box-schemes with finite-differencing interpretation, [Ke 71], [Co 92], the degrees of freedom are located at the center of the faces of the mesh. Nevertheless, the discretization is interpreted here as a Finite Element approximation, allowing to use Finite Element theory for the numerical analysis. Let us consider the 2D Poisson problem in mixed form with Dirichlet homogeneous boundary conditions
Suppose given a triangulation Th of the domain ft C R2 by triangles K. The number of triangles is NE. The number of internal edges, boundary edges are NAi, NAb and the total number of edges is NA = NAi + NAb. The Finite Element spaces that are used are u-space: The non-conforming Crouzeix-Raviart space with homegeneous boundary conditions Vh = P^c 0, equipped with the mesh dependent norm
Numerical analysis
241
p-space: The div-conforming Raviart-Thomas space of least order Qh = RT°, equipped with the continuous norm
Recall that these spaces are Vh — {^/i/V K 6 Th, Vh\K £ Pl(K},Vh is continuous at the middle of each edge,^ = 0 at the middle of each edge on <9f7)
The scheme reads: find (w^p/J e Vh x Qh such that
In (2), the number of unknowns is IN A, since the global degrees of freedom for Uh, Ph are scalars located at the center of the faces of the mesh. The number of equations is clearly SNE + NAf,. A simple count of the faces (the edges) proves that in fact we have
Let us mention that coupling these two spaces is not standard in the mixed finite element methods, because they do not verify the Babuska-Brezzi condition, [Ba 71], [Bre 74]. 3. Numerical Analysis
3.1. Reformulation as a mixed Petrov-Galerkin method We consider the following mixed formulation of problem (1): find (u,p) £ HQ x #div such that for any (v,q) € L2 x (I/ 2 ) 2
or equivalently
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Finite volumes for complex applications
Applying the general Babuska theorem [Ba 71] onto mixed formulation we find easily that (5) is a well posed problem, whose solution is (u, Vw) , u G HQ r\H2 being the unique solution of the original problem (1). The scheme (2) appears now as a Petrov-Galerkin non conforming approximation of (5). Calling P° the space of constant functions in each triangle, it can be rewritten: find (uh,ph) & P^C]0 x RT° such that for any (vh,qh) e P° x (P°)2
or equivalently
Applying now the theory for mixed Petrov-Galerkin approximations, [Ni 82], [BCM 88], [BMO 96], [Cr 99], we get the following result: Theorem 1 (i) The scheme (7) does possess an unique solution (uh,ph) £ P^o x RT° verifying
(ii) We have the error estimate
3.2. Dual scheme Another mixed form of the Poisson problem linked with the bilinear form B is dual from (1): find (v, q) G L2 x (L2)2 such that for any (u,p) e HQ x H<&v
or equivalently
Numerical analysis
243
Again by the Babuska theorem, problem (11) does possess an unique solution (v,q) = (u, Vw). The corresponding Petrov-Galerkin scheme is: find (vf^Qh) € P° x (P°)2 such that
As precedingly, we get the following result Theorem 2 (i) The dual scheme (12) does have an unique solution (t>/n
(ii) We have the error estimate
Note that (12) defines a non standard cell-centered Finite-Volume scheme for computing both the unknown and the gradient from the knowledge of the laplacian. 3.3. Second order error estimate Using the error estimates for the primal and dual schemes (7), (12), allows to derive a second order error estimate in the L2 norm for the unknown Uh in (7). The proof uses an Aubin-Nitsche like argument. Theorem 3 The solution (uh,Ph) °f Scheme (7) verifies the optimal error estimate
3.4. Further remarks A natural question is to ask whether there is the link between this scheme and the family of Finite Element mixed methods, [RT 77], [AB 85], [BDM 85], [BF 91]. In fact, it can be proved that the gradient part ph in (7) coincides with the mixed gradient p^ in [RT 77]. However, the Uh part in the mixed method
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Finite volumes for complex applications
is only a P° approximation of u. The optimal error estimate is therefore only of first order in the L2 norm. They are several methods in order to interpolate a posteriori Uh to an higher order approximation, [AB 85]. All these methods introduce a three variables problem (uh->ph, ^h), involving a new degree of freedom (a Lagrange parameter) A/,, at the interfaces of the mesh. It can be proved, [Cr 99], that one of these interpolations coincides in fact with the scheme (7). Moreover, contrary to the standard mixed method in its original formulation, the degrees of freedom for Uh and ph are decoupled, allowing to solve only an O(NA) system in Uh-, which is in addition symmetric definite positive. We refer to [CoC 98] for details onto the implementation of (7), and to [Cr 99] for the proofs of the results described here. 4. Conclusion Several works are devoted to the a posteriori interpretation of mixed Finite Element methods as Finite Volume ones for the primal unknown Uh- In other works, they are attempts to compute more easily this unknown, by introducing additional degrees of freedom. The advantage of the scheme (7) is that it is basically designed as a true Finite Volume scheme on a single computational cell, for Uh and ph. In addition, it gives a natural decoupling between the unknowns Uh and ph. Moreover, it has an optimal order of accuracy both for Uh and ph, without any post-processing or a posteriori interpretation. Note that this kind of schemes is not restricted to triangular meshes, or to the dimension 2. We think that the formulation of this scheme as a Petrov-Galerkin method, combining the advantages of mixed and Finite Volume methods, can be particularly interesting for computations involving complex fluxes. Moreover, it can be of some help for a better understanding of the link between mixed and cell-centered Finite Volume methods. Higher order extensions are currently in progress. Acknowledgments: The author acknowledges friendly B. Courbet, F. Dubois, and A. Debussche, J. Laminie, for stimulating discussions and encouragements.
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References
[AABM 98] B. ACHCHAB, A. AGOUZAL, J. BARANGER, J-F. MAITRE, Estimateur d'erreur a posteriori hierarchique. Application aux elements finis mixtes Numer. Math. 80, 1998, 159-179. [AB 85] D.N. ARNOLD, F. BREZZI, Mixed and non-conforming finite elements methods: implementation, postprocessing and error estimates, Math. Model, and Numer. Anal. 19,1, 1985, 7-32. [Ba 71] I. BABUSKA, Error-Bounds for Finite Elements Method, Numer.Math., 16, 322-333. [BR 87] R.E. BANK, D.J. ROSE, Some error estimates for the box method, SIAM J. Numer. Anal, 24,4, 1987, 777-787. [BMO 96] J. BARANGER, J.F. MAITRE, F. OUDIN, Connection between finite volume and mixed finite element methods, Math. Model, and Numer. Anal., 30,4, 1996, 445-465. [BCM 88] C. BERNARDI, C. CANUTO, Y. MADAY, Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem SIAM J. Numer. Anal., 25,6, 1988, 1237-1271. [Bre 74] F. BREZZI, On the existence, uniqueness and approximation of saddlepoint problems, arising from lagrangian multipliers R.A.I.R.O. 8, 1974, R-2, 129-151. [BF 91] F. BREZZI, M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer Series in Comp. Math., 15, Springer Verlag, New-York, 1991. [BDM 85] F. BREZZI, J. DOUGLAS, L.D. MARINI, Two families of Mixed Finite Element for second order elliptic problems, Numer. Math., 47, (1985), 217-235. [CMM 91] Z. CAI, J. MANDEL, S. McCoRMiCK, The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal., 28,2, 1991, 392-402. [CDH 83] F. CASIER, H. DECONNINCK, C. HIRSCH, A class of central bidiagonal schemes with implicit boundary conditions for the solution of Euler's equations, AIAA-83-0126, 1983. [CM 86] J.J. CHATTOT, S. MALET, A "box-scheme" for the Euler equations, Lecture Notes in Math., 1270, Springer-Verlag, 1986, 52-63.
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[Co 92] B. COURBET, Schemas boite en reseau triangulaire, Rapport technique 18/3446 EN, (1992), ONERA, unpublished. [Co 91] B. COURBET, Etude d'une famille de schemas boites a deux points et application a la dynamique des gaz monodimensionnelle, La Recherche Aerospatiale, n° 5, 1991, 31-44. [CoC 98] B. COURBET, J.P. CROISILLE, Finite Volume Box Schemes on triangular meshes, Math. Model. and Numer., 32,5, (1998), 631-649. [Cr 99] J-P. CROISILLE, Finite Volume Box Schemes and Mixed Methods, Preprint, Universite de Metz, 1999. [Du 97] F. DUBOIS, Finite volumes and mixed Petrov-Galerkin finite elements; the unidimensional problem. Preprint 295 du C.N.A.M., 1997. [EGH 97] R. EYMARD, T. GALLOUET, R. HERBIN, Finite Volume Methods, in Handbook of Numerical Analysis, vol. 5, Ciarlet-Lions eds., (1997). [FS 89] L. FEZOUI, B. STOUFFLET, A class of implicit upwind schemes for Euler equations on unstructured grids, Jour. of Comp. Phys., 84, 1989, 174-206. [Ha 89] W. HACKBUSCH, On first and second order box schemes, Computing, 41, 1989, 277-296. [Ke 71] H.B. KELLER, A new difference scheme for parabolic problems, Numerical solutions of partial differential equations, II, B. Hubbard ed., Academic Press, New-York, 1971, 327-350. [Ni 82] R.A. NICOLAIDES, Existence, uniqueness and approximation for generalized saddle point problems, SIAM J. Numer. Anal., 19,2, 1982, 349-357. [RT 77] P.A. RAVIART, J.M. THOMAS, A mixed finite element method for 2nd order elliptic problems, Lecture Notes in Math., 606, SpringerVerlag, 1977, 292-315. [TS 93] T. SCHMIDT, Box Schemes on quadrilateral meshes, Computing, 51 1993, 271-292. [YMAC 99] A. YOUNES, R. MOSE, P. ACKERER, G. CHAVENT, A new formulation or the Mixed Finite Element Method for solving elliptic and parabolic PDE, Jour. of Comp. Phys., 149, 1999, 148-167.
On nonlinear stability analysis for finite volume schemes, plane wave instability and carbuncle phenomena explanation
M. Abouziarov Research Institute of Mechanics Loba chevskey university, Nizhnii Novgorod, Russia e-mail [email protected]
1. Introduction
Courant restriction of integration time step, which follows from stability analysis of linearised Euler equations, does not provide the stability of some numerical methods for nonlinear problems . Some of these problems and attempt of analysis of the origin of these instabilities were made in [1]. In this report a procedure of the instability analysis for finite volume methods for the nonlinear equations and a method to correct the numerical schemes to avoid these instabilities are presented.
2. Nonlinear stability analysis
Assume for a 2D case a cell number (i,j) is integrated, where integer indexes i, j ( X and Y direction respectively ) are related with parameters (i.e. grid, pressure, density, velocities, energy) at the cell centers and half-
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Finite volumes for complex applications
pressure and velocity perturbations are small, but there can be the density discontinuity. For the calculation process to be stable the perturbation of the new velocity is to be equal to the old one or this velocity perturbation is to be diminished step by step.
Assume:
respectively
and
and etc.
(values with covers are perturbations of the appropriate values). For the original Godunov?s method, using linearised Riemann?s solution with the different left and right densities and sonic velocities, the following simple estimation can be obtained:
A A similar formula is for unew For the second order modification [3], the formulas
are analogous. For linearised Euler equations, when
, it is
obvious from (6) that the Courant restriction is enough for stability. But when there is the density discontinuity and the change of the density in the cell during one step is big, to provide the calculation stability the time step is to be stronger restricted or the scheme is to be reconstructed to correct these additional instability terms; without it even small perturbations of the velocity may to increase. It can be shown that on the back of the shock wave instability estimations are similar (6) but with stronger possible instability. For example, such instability situations are possible on the back of a strong shock wave in a wind tunnel (plane wave instability) or near the head of the shock wave, generated by the sphere or other body in the supersonic flow ?carbuncle phenomena?. In the ?carbuncle phenomena?, at the beginning of the iterations, the situation is very similar to the plane wave instability case - a strong
Numerical analysis integer indexes show the parameters on the cell boundaries.
Eq.l represents the mass conservation law for this cell. For simplification (without any restrictions) the grid is assumed to be stationary and regular. °' '
are
respectively volume , surface area of a boundary of this cell and integration time step; u and v are velocities for X and Y directions respectively; and the values with the upper indexes indicate the new time step level.
Eq.2 represents the momentum conservation law in Y direction for this cell (similar for X direction). Using the expression for
ot
^J from eq. 1, it follows from Eq.2 ,
Eq.4 is a formula for obtaining the velocity on the new time step based on the velocity for the previous one. The formulas for velocity in the X direction and for pressure derived from the energy equation have a similar form. These formulas are nonlinear and follow from the conservation laws. Using different ways for obtaining the fluxes on the cell boundaries, different numerical schemes can be constructed. Let us consider the behavior of (4) for the plane wave in the X direction in the vicinity of the density discontinuity for widely used Godunov?s method. Assume that our flow is near to equilibrium in its pressure and velocity and the values of
249
250
Finite volumes for complex applications
shock wave is separated from the body and goes ahead.
3. A possible way of correction
A possible way to correct this instability follows from formula (6): it is necessary to eliminate the perturbations which appear in addition to the standard Courant perturbations. These corrections should have the following logic: corrections are to have higher order accuracy than that of the numerical scheme, they are not to change the approximation error; they are to correct only the instability terms to avoid the increase of the perturbations. One way to make the scheme stable is to use appropriate corrections of instability terms after integration, but in this case it is necessary to analyze the other kind of discontinuity problem more accurate (shock wave) too, which is rather difficult. The second way to eliminate instability terms is to correct such terms in the fluxes [2]. For the momentum equations (3) the correction term is as follows:
where the values with the covers (respective errors) are to have the second order accuracy in the appropriate directions. For example:
Numerical analysis
The momentum fluxes for the X direction and energy fluxes are corrected in a similar way. With these corrections, the original Godunov method and its second order accuracy modification [3] become stable with standard Courant stability restrictions for those schemes and do not have the plane wave instability and ?carbuncle phenomena? troubles. In (5) it was assumed that near the pressure and velocity equilibrium state the absolute values of the appropriate perturbations are the same for different cells; it is reasonable because the relative truncation error for all values is the same and depends only on the type of a computer. From our density discontinuity analysis it follows that the velocity and pressure perturbations do not compensate each other after integration in formula (4) because the flux perturbations are proportional to the appropriate densities, which are different. For a shock wave; the situation is more complicated - absolute perturbation values are different and are related with the absolute values of the respective fields.
4. Conclusions
Thus, it may be concluded that: in constructing numerical methods and analyzing their stability for computer calculations in the regions of possible big gradients, it is necessary to take into account that only relative truncation error is the same for all numbers in the computer, whereas absolute values of perturbations are proportional to this relative truncation error and to those absolute values.
5. Bibliography
[1] J. QUIRK, "A contribution to the great Riemann solver debate," International Journal for numerical methods in fluids 18 (1994), pp. 555-574. [2] ABOUZIAROV M. Nonlinear stability analysis of Euler equations for predictor corrector schemes. // Godunov?s method for gas dynamics: Current Applications and Future
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Finite volumes for complex applications
Developments, A Symposium Honoring S.K. Godunov, May 1-2, 1997, The University of Michigan, Ann Arbor, Michigan [3] ABOUZIAROV M. On accuracy increasing of Godunov's method for nonlinear problems of continuum mechanics. // Godunov?s method for gas dynamics: Current Applications and Future Developments, A Symposium Honoring S.K. Godunov, May 1-2, 1997, The University of Michigan, Ann Arbor, Michigan
Innovative schemes
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A comparison between upwind and multidimensional upwind schemes for unsteady flow
P. Brufau and P. Garcia-Navarro Centra Politecnico Superior 50015 Zaragoza, Spain
ABSTRACT An ideal scheme for the solution of multidimensional non-linear systems of partial differential equations governing fluid motion has not yet be found despite years of research effort and many scientific contributions. In the context of finite volume approximations based on flow dependent schemes, discussion is opened as to whether upwind or multidimensional upwind schemes are preferable for solving 2D problems. In this work we consider the use of upwind and multidimensional upwinding techniques for 2D shallow water flows, in particular, to the simulation of dam break flows. The basis of the two numerical methods is stated and the particular adaptation to the shallow water system is described. As test cases, laboratory experimental data supplied by partners of the Working Group on Dam Break Flow Modelling (CADAM) are used. Key Words: upwind schemes, shallow water, dam break.
1. Introduction Upwind schemes[l, 9] are accepted as accurate methods for the numerical solution of systems of conservation laws in one dimension. Attempts of applying these techniques in higher dimensions have generally relied on essentially one-dimensional algorithms combined with some form of operator splitting. Indeed, the standard extension of the finite volume methods is still to solve onedimensional Riemann problems created at the interfaces of the finite volumes by the discontinuities in the reconstruction stage of the algorithm. When extended in this way many features of the flow, particularly if they are not aligned with the grid, can be completely misinterpreted by the numerical scheme. Discussion is open as to whether the schemes based on ID Riemann solvers
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Finite volumes for complex applications
are the most suitable choice for multidimensional calculations because they seem inadequate for capturing 2D flow features. Recently, a class of upwind methods has emerged which attempts to model the equations in a genuinely multidimensional manner[2, 3, 5, 6]. These schemes are designed to monitor the average time evolution of the approximation to the solution within a complete grid cell rather than concentrating on the activity at the interfaces. Multidimensional upwind schemes were developed initially for the approximation of steady state solutions of the two-dimensional Euler equations on unstructured triangular grids, although they could be applicable to any system of hyperbolic conservation laws. One such case is given by the shallow water equations[4]. In this paper the performance of an upwind and a multidimensional upwind technique for 2D shallow water flows is described in first order accuracy. In the next sections, the basis of the numerical methods is stated and the application to the simulation of 2D dam break flows is presented. 2. Governing equations
Depth averaging of the free surface flow equations under the shallow water hypothesis leads to a common version of the 2D shallow water equations which, in conservative form is,
with
where U represents the vector of conserved variables (h depth of water, hu and hv unit discharges along the coordinate directions), F and G are the fluxes of the conserved variables across the edges of a control volume. They consist of the convective fluxes together with the hydrostatic pressure gradients. H is the source term. In addition, u and v are the velocities in the x, y directions respectively, g is the acceleration due to the gravity, SQX, SQy are the bed slopes and Sfx, Sfy the friction terms in the x, y directions. For the friction term, the Manning equation has been used.
3. Numerical models
3.1 Upwind method A cell centered finite volume method is formulated for equation (1) over a triangular control volume where the dependent variables of the system are represented as piecewise constants. The integral form of (1) for a fixed area 5 is
and, applying the divergence theorem to the second integral, we obtain
where C is the boundary of the area 5, and n is the outward unitary normal vector. Given a computational mesh defined by the cells (volumes) of area Si, where i is the index associated with the centroid of the cell in which the cellwise constant values of U are stored, equation (4) can be represented by
where a mesh fixed in time is assumed. The contour integral is approached via a mid-point rule, i.e., a numerical flux is defined at the mid-point of each edge, giving
where Wk represents the index of edge k of the cell, NE is the total number of edges in the cell. The vector nwk is the unit outward normal, dCWk is the length of the side, and (F, G)^ fc is the numerical flux tensor. The evaluation of the numerical flux in equation (6) is based on the Riemann problem defined by the conditions on the left and right sides of the cell edges. An important feature of the 1D upwind schemes for non-linear systems of equations is exploited here. This is the definition of the approximated flux jacobian, Ai+i, constructed at the edges of the cells and satisfying special conditions[7]. The 2D numerical upwind flux in equation (6) is obtained by applying the expression of the numerical flux across the interface i + \ of a cell in a 1D domain to each edge Wk of the computational cell in a 1D form. The ID philosophy is followed along the normal direction to the cell walls, making use of the normal numerical fluxes, so that
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Finite volumes for complex applications
R and L denote right and left states respectively at the Wk edge, (ARL) represents the approximate Jacobian of the normal flux. As suggested by Roe[7] the matrix ARL has the same shape as A, being
but is evaluated at an average state[l]. We can now substitute (7) into (6), so that (5) can be written as
which is an ordinary differential equation and can be integrated by standard methods such as a forward Euler time integration procedure.
The stability criterion adopted has followed the usual in explicit finite volumes [1]
where dij is the distance between the centroid of the cell i and its neighbours. 3.2. Multidimensional upwind method The application of this technique to the shallow water system requires the equations to be written in terms of the conserved variables, so that (1) can be expressed in its non-conservative form as
in which only the homogeneous part has been considered. It is useful to express the equations in terms of the primitive variables in a non-conservative way, as
In the conservative formulation, the fluctuation is defined as
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We can use the relation between the two sets of variables to define new matrices R and S,
so that Provided that the variables V are linear over the cells T, the gradients, Va and Vy, are constant, and we can write the fluctuation as with the definitions:
We approximate R, S by where the averaged variables are simply
summing over the nodal values at the vertices of the triangle T. Note that with this definition of R, S we are only approximating equation (18). As a consequence, we lose conservation in the numerical evaluation of the fluctuation. The next step consists on computing the fluctuations and distributing them to the vertices of every cell by means of an advection scheme[3]. This last step requires the description of wave models[8] and the choice of an advection scheme. 4. Numerical results Results obtained with first order upwind and multidimensional upwind approximations on unstructured Delaunay triangular meshes for the experimental test case proposed by Prof. Zech (Civil Engineering Dpt., UCL Belgium) from the Working Group on Dam Break Flow Modelling are going to be presented. The test combines a square shaped upstream reservoir and a 45° bend channel (see Fig. 1). The flow will be essentially two-dimensional in the reservoir and at the angle between the two straight reaches of the 45° bend channel. Two features of the dam break resulting flow are of special interest: the damping effect of the corner, and the upstream moving hydraulic jump which is formed by reflection at the corner.
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The initial conditions are water at rest with the free surface 25cm in the upstream reservoir and 1cm water depth in the channel. All boundaries are solid non-slip walls except the outlet which is considered free. The Manning coefficient is n^ = 0.0095 for the bed. The number of elements used in the mesh is 15397. Nine gauging points were used in the laboratory to measure water level in time. Their location is shown in Fig. 1. The measurements at these stations are compared to the numerical results and displayed in Fig. 2. In Fig. 3 the free surface is described at time 10s and finally, Fig. 4 represents snapshots of the free surface at time 18s. In general, the figures indicate a good performance of the two numerical schemes. The arrival times of the main shock fronts is better captured by the upwind method. Some differences are noticeable in P3 and P4 as the reflected shock front celerity is concerned. This may be attributed to the treatment of the boundary conditions. The great difference between the results obtained with both schemes is referred to the free surface plot. Perhaps measurements along the walls of the channel should be taken into account to demonstrate which approximation is better. Till now, only data in the central axis have been measured. 5. Conclusions
An upwind and a multidimensional upwind scheme for the solution of the 2D shallow water equations has been applied in first order accuracy for dam break modelling. The numerical results have been validated by comparison with experimental data in one test case. Differences on the results obtained with both techniques do not follow a clear tendency and it is difficult to establish the superiority of one over another. In the test case presented, both results fail to reproduce exactly the arrival times of the reflected wave (P2 and P3 gauge points). This suggests that the reflection at the corner may require an improved numerical treatment. Moreover, differences can be noticed at the free surface plots. Reflected waves from the walls are clearly observed in the multidimensional upwind results and perhaps a good test could be performed taking measures along the channel walls to give a new point of view in the numerical analysis.
6. Bibliography
[1]
[2]
BRUFAU P., An upwind scheme for the 2D shallow water equations with applications , Num. Anal. Rep. 11/97, University of Reading, England (1997).
BRUFAU P. AND GARCIA-NAVARRO P., Two dimensional dam break flows in unstructured grids, Hydroinformatics98,
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Copenhague, Denmark (1998). [3]
DECONINCK H. et al., Multidimensional Upwind Methods for Unstructured Grids, Unst. Grid Met. for Advec. Domin. Flows, AGARD, 787(1992).
[4]
GARCIA-NAVARRO P. et al., Genuinely Multidimensional Upwinding for the 2D Shallow Water Equations, J. Comp. Phys., 121 (1995), p. 79-93.
[5]
HUBBARD M.E., Multidimensional Upwinding and Grid Adaptation for Conservation Laws, PhD Thesis, University of Reading, England, 1996.
[6]
PAILLERE H., Multidimensional Upwind Residual Distribution Schemes for the Euler and Navier-Stokes Equations on Unstructured Grids, PhD Thesis, University of Brussels, Belgium, 1995.
[7]
ROE P.L., A Basis for Upwind Differencing of the TwoDimensional Unsteady Euler Equations, Num. Met. Fluid Dyn. II (1986).
[8]
RUDGYARD M., Multidimensional Wave Decompositions for the Euler Equations, VKI Lecture Series, Comput. Fluid Dynam. (1993).
[9]
SLEIGH P.A. et al., An Unstructured Finite Volume Algorithm for predicting flow in rivers and estuaries, Comp. and Fluids (1997).
7. Figures
Figure 1: Plane view of the channel.
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Figure 2: Water depth history at points a) P3, b) P4, c) P5 and d) P9.
Figure 3: Free surface at time t=10s with upwind (a) and multidimensional upwind schemes (b).
Reformulation of the unstructured staggered mesh method as a classic finite volume method Blair Perot & Xing Zhang University of Massachusetts 219 Engineering Laboratory Amherst, MA 01003
ABSTRACT A generalization of the Harlow & Welch (1965) staggered mesh method to twodimensional unstructured meshes is presented. With certain choices of the interpolation operators, it is shown that this method can be recast as a classic finite volume method using a single set of non-overlapping control volumes and collocated variables. When the divergence form of the Navier-Stokes equations are discretized using the unstructured staggered mesh method the resulting equations are equivalent to a classic finite volume method for the velocity vector. When the rotational form of the Navier-Stokes equations are discretized using the unstructured staggered mesh method the resulting equations are equivalent to a classic finite volume method for the vorticity vector. Key Words: unstructured, staggered mesh, reformulation, Navier-Stokes equations.
1. Introduction The Cartesian staggered mesh method has a number of mathematical properties that make it a popular choice for simulations of incompressible fluids. In particular, the method does not have spurious 'pressure modes' and does not require stabilization or damping terms to control unphysical small-scale pressure fluctuations. In addition, the method is known to conserve mass, momentum, total energy, kinetic energy and vorticity. The latter two conservation properties are not found in generic control volume approaches and are particularly important in direct and large eddy simulations of turbulence where the cascade of turbulent kinetic energy (or enstrophy) from large to small scales (or vice versa) is critical to the overall predictions of the turbulence behavior.
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The success of the Cartesian staggered mesh method originally developed by Harlow & Welch [HAR 65] has motivated the search for generalizations of the method to unstructured meshes. While such a generalization is a non-trivial exercise, the unstructured staggered mesh methods of Porsching [AMI 81], and Nicolaides [NIC 93] have demonstrated many of the attractive properties of the Cartesian staggered mesh method by taking advantage of the fact that every unstructured mesh has a locally orthogonal dual mesh - the Voronoi tesselation. Chou [CHO 97] has shown the connection of the unstructured staggered mesh methods to nonconforming finite element methods. In this paper we discuss the direct connection with classic finite volume methods. It is via this connection with classic finite volume methods that the hitherto uninvestigated conservation properties of unstructured staggered mesh methods can be evaluated.
Figure 1. Two dimensional unstructured mesh and the dual Voronoi mesh.
2. Analysis of the Divergence Form The unstructured staggered mesh discretization is simply a way of forming discrete difference operators. It is actually independent of the equations to which it is applied. Hence, different discretizations of the Navier-Stokes equations are possible depending on which form of the equations are discretized. In this section, we will look at unstructured staggered mesh discretizations of the divergence form of the Navier-Stokes equations.
Discretizations based on the divergence form of the equations are of interest because they are able to discretely conserve momentum. While momentum conservation is a trivial consequence of a classical finite volume method, it is not an obvious trait of staggered mesh methods. This is because the staggered mesh methods only updates the normal velocity components at cell faces, tangential
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velocity components are interpolated not evolved. It will be shown that with certain choices of the interpolation operators, the staggered mesh update of face normal velocities is directly equivalent to a classic finite volume method which updates the velocity vector at cell centers. 2.1 Discretization of the Divergence Form The normal vector at each face is assumed to point from cell Cl to cell C2. At boundary faces the normal vector is assumed to point out of the domain and cell C2 is a virtual cell located at the domain boundary. The discrete equation for the evolution of the normal velocity component is then given by,
where c t = - ^ - £ u f u A f
is a conservative discretization of the convection term cell faces
evaluated
in each
cell,
dc =^- !>(Vu + Vu ) - n f Af
is a conservative
discretization of the diffusion term evaluated in each cell, V. is the volume of each cell, Af is the face area, Wf is the distance between neighboring cell circumcenters, and W^ is the distance between the face circumcenter and the cell circumcenter. Note that u = u • nf is the normal velocity component at each cell face and u is the normal velocity component that points out of a particular cell. Similarly nf is the normal vector pointing out of a particular cell. 2.2 Reformulation as a Classic Control Volume Scheme The reformulation of the divergence form is accomplished by multiplying each evolution equation for the face normal velocity component (Eqn. [2]), by the face normal vector, and summing over all the faces in the computational domain. This results in the following equation,
The goal is then to recast this into a form that looks like a summation over control volume cells. Recognizing that Wf = W^ + W^ and that W^ = 0 at
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boundary faces, and also noting that at boundary faces pc2 = pf, rewritten as a summation over cells.
[3] can be
This can be further simplified using three identities derived from Gauss' divergence theorem. Gauss' Divergence Theorem for an arbitrary bounded volume and a vector quantity f is,
where Q is the volume and dQ, is the boundary of the volume with unit normal vector n oriented outwards from the volume. We are actually interested in convex polygonal volumes where Gauss' Theorem simplifies to,
cell faces
If f is a nonzero constant vector then £ nf Af = 0 . If f = (x • a)b where a and b are nonzero constant vectors and x is the position vector with an origin located at the cell circumcenter then it follows from [6] that in two-dimensions cell faces
X n r n f W c f A = VCI where I is the identity matrix. Finally, if f = (a-x)u where u
is the velocity vector and a is an arbitrary nonzero constant vector, then Gauss' Theorem gives,
The gradient of the position vector is the identity matrix ( x s . = 5sj), and since a is an arbitrary vector
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where u is the outwards normal component of the velocity at the cell faces. This is an exact equation for polygonal volumes. If we assume that the velocity field u is a constant function (a first order approximation), then the second term will be zero and the integrals can be evaluated. In two dimensions, this gives the relation,
The first two identities are geometric and they are exact. The last expression (Eqn. [9]) is really not an identity, it is a first order approximation for the cell velocity vector given the normal velocity components at the cell faces. With these geometric identities and this definition for the cell velocity vector, [4] becomes,
This equation is true for a collection of cells, but it is also true for a single mesh cell. The preceding analysis makes no distinction as to the number of cells. Applying the previous definitions for cc and dc we can therefore write that
This is true for each mesh cell, and has the form of a classic control volume scheme for the velocity vector in the mesh cells. It is important to note however, that despite the apparent similarity there remains a subtle distinction from classic control volume schemes. In the staggered mesh scheme, the normal velocity component u is the primary unknown and uc is a derived quantity. In classic control volume schemes, uc is the primary velocity unknown and the normal velocity component at faces is derived. 3. Analysis of the Rotational Form In this section, we will look at unstructured staggered mesh discretizations of the rotational form of the Navier-Stokes equations.
where u is the velocity vector, co is the vorticity, pd = p + y u - u is the specific dynamic pressure, and v is the kinematic viscosity. This equation assumes that viscosity is constant, but it is otherwise equivalent to other forms of the incompressible Navier-Stokes equations. Variable viscosity diffusion can be still be
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represented in rotational form but the extra term (involving second derivatives of viscosity) complicates the analysis unnecessarily. This particular form of the Navier-Stokes equations is of interest because it appears to be inherently suited to the staggered mesh discretization. The classic staggered mesh method can be rearranged to look like a discretization of [12]. It will be shown that in two dimensions the staggered mesh update of face normal velocities is directly Figure 2. Notation for a cell face in relation equivalent to a classic finite to neighboring cells and nodes. volume method which updates the vorticity at nodes and where the control volumes are the dual mesh Veronoi polyhedra. 3.1 Discretization of the Rotational Form Using the rotational form of the Navier-Stokes equations, the normal component of the face velocity is discretized as,
where con is the vorticity at a node in the direction out of the two-dimensional plane. The face tangential points from node Nl to node N2 and is oriented 90 degrees counterclockwise to the face normal vector. The tangential velocity at the nodes in the convection term is given by v n = u n • t f . 3.2 Reformulation as a Control Volume for Vorticity The reformulation of the rotational form is accomplished by dividing each normal velocity evolution equation (Eqn. [13]) by the face area and then multiplying by -1 if the face normal points clockwise with respect to the node in question, and finally summing over all the faces touching a specific node. The result will be shown to be a control volume equation for the vorticity. In mathematical notation we start with the following equation,
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where the normal vector at each face has been chosen to point in a direction counterclockwise with respect to the node in question. In addition, node n0 is the node around which the summation is occurring and node ni is the other node adjoining that face. In this case, we use a discrete version of Stokes Curl Theorem to simplify the equations. Stokes theorem says that for an arbitrary bounded surface and vector quantity f,
where S is the surface with normal z, 3S is the boundary- of the surface, and the integration takes place in a counterclockwise direction around the boundary with respect to the face normal. We are actually interested in the planar polygonal Veronio regions surrounding each node, it which case Stokes Theorem simplifies to
If we let f equal the velocity vector and make the first order assumption that the velocity is constant in the Veronio cell then we obtain,
where An is the area of the Veronio cell surrounding the node, and the face normal vectors are assumed to point in a counterclockwise direction around the node. In conjunction with [17] it is clear that for interior nodes, the pressure term is identically zero. The net result is that [14] can be written as,
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where the face vorticity flux is given by (vco) lf = T(co n0 u n0 + co nj u ni ) • tf and tf points out of the Veronio cell. This is a discrete version of the continuous two-dimensional vorticity evolution equation,
Again, it is important to note that despite the apparent similarity there remains a subtle distinction from classic control volume schemes. In the staggered mesh scheme, the normal velocity component u is the primary unknown and con is the derived quantity. In a classic control volume schemes, con would be the primary unknown. So the staggered mesh scheme differs from a standard vorticitystreamfunction or vorticity-velocity formulation in the fact that (often complex) boundary conditions on the vorticity are not required. 4. Discussion The primary result of the current work is that staggered mesh methods are not just control volume methods applied on staggered control volumes, but are directly equivalent to classic collocated control volume methods. It was shown that unstructured staggered mesh discretizations of the divergence form of the NavierStokes equations are equivalent to classic control volume method for the velocity vector in mesh cells. Likewise, unstructured staggered mesh discretizations of the rotational form of the Navier-Stokes equations are equivalent to classic control volume method for the vorticity vector at mesh nodes (in Veronio cells). These equivalencies imply that the method possesses certain conservation properties. References [HAR 65]
HARLOW, F. H. & WELCH, J. E., Numerical calculations of time dependent viscous incompressible flow of fluid with a free surface, Phys. Fluids, 8 , 1965, p. 2182.
[NIC 93]
NICOLAIDES, R. A., The covolume approach to computing incompressible flow, Incompressible Computational Fluid Dynamics, M. D. Gunzburger & R. A. Nicolaides, eds., Cambridge University Press, 1993, p. 295-234.
[AMI 81]
AMIT, R., HALL, C. A. & PORSHING, T. A., An Application of Network Theory to the Solution of Implicit Navier-Stokes Difference Equations, J. Comput. Phys. 40, 1981, p. 183.
[CHO 97]
CHOU, S. H., Analysis and convergence of a covolume method for generalized Stokes problem, Math, of Comput. 66 (217), 1997, p. 85.
A mixed FE - FV algorithm in non-linear solid dynamics
Serguei' V. Potapov EDF, Research and Development Division Acoustics & Vibration Mechanics Branch 1, avenue du General-de-Gaulle, B.P. 408 92141 Clamart Cedex France
ABSTRACT. This article presents an extension to the transient non-linear solid dynamics of a mixed Finite Element-Finite Volume algorithm initially proposed for simulation of hydrodynamic problems with compressible fluids. This algorithm is based on an Arbitrary Lagrangian Eulerian (ALE) formulation of motion. By using a fractional step method, the ALE problem is divided in two phases : a Lagrangian one and a transport one. The Lagrangian phase dealing with spatially symmetric terms is approximated using a conventional Galerkin Finite Element formulation. To deal with the transport phase, whose operator is not symmetric, the Finite Volume formulation is applied. Using two kinds of finite volume cells warrants a compatibility of spatial approximations of the Lagrangian and transport phases and allows to perform correctly the transport of all unknown variables. Key Words : non-linear solid dynamics, finite element, finite volume, explicit scheme
1. Introduction The ALE formulation is well established in fluid dynamics where it is applied to the transient analysis of flows on moving meshes. The capacity of the ALE kinematics to manage strong fluid flows on dynamically deforming meshes is attractive to use in the non-linear solid mechanics dealing with large elastic-plastic deformations. However, this extension is not trivial because the stress tensor of solids is not only function of instantaneous quantities (as for fluids) but depends on the specific history of each material point. In order to evaluate the stress state on a current step, the constitutive equation must be integrated in time. Since in the ALE formulation
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the computational grid and the material move independently, the quadrature points at which the stresses are evaluated coincide throughout the deformation process with different material points, having in general different deformation histories. Thus, the stress transport procedure is needed to overcome the numerical difficulties due to relative motion between grid nodes and material particles. Furthermore, the same update procedure must be applied to each internal variable of the elasto-plastic analysis, e.g. to the yield stress in isotropic hardening or to the back stress in kinematic hardening. Direct time integration of the ALE equations is not easy because they contain the spatial derivative terms corresponding to very different physical phenomena. So, we use a fractional step method to integrate in time such equations. The ALE problem of each time step is split into two phases. The first one is so called Lagrangian phase, in which the computational grid is assumed to follow the material points, what cancels all transport terms. The second phase is a convective one in which only the transport terms are taken into account. The resolution of ALE problems on unstructured grids is generally made via the weak formulation suited by the finite element approximation. However, the standard Galerkin finite element approach produces spurious numerical oscillations when applied to non-symmetric operator of the transport phase. Therefore, the upwind technique must be used to overcome these instabilities. In [HUE 95], two distinct strategies, borrowed from the fluid dynamics experience, are presented. A Lax-Wendroff scheme and a Godunov algorithm are implemented in finite element context to perform the transport of stresses and stress-related variables. To stabilise the momentum equation some ad-hoc donor-element upwind technique is used. Here, we use the finite volume technique to perform, in unified and conservative way, the convection of all unknown variables [POT 97]. 2. A mixed approach using Finite Element and Finite Volume formulations First, we rewrite the standard ALE equations under a conservative way reinforcing the momentum, the energy and the solid time dependent constitutive equations by the mass conservation equation. Then, the ALE system is split into two sub-systems. All symmetric terms like local strains of the continuum and diffusion terms are grouped in the first sub-system which corresponds to the Lagrangian phase. The second subsystem contains only purely convective terms describing the motion of the material through the ALE computational grid. It corresponds to the transport phase. Such a splitting is very general and allows to consider both geometric and material nonlinearities related to the problem. Furthermore, it enables us to choose for each phase the appropriate integral formulation and method of resolution. 2.1 Finite Element formulation of the Lagrangian phase The equations of the Lagrangian phase possess a self-adjoint (symmetric) spatial operator. To guarantee the best approximation, we apply to thein a conventional Galerkin integral formulation with symmetric weigh functions Wp,Wv,We. Due to the fact that all equations of the ALE formulation contain convective terms, we must
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consider the mass density p, the material velocities t>,, the internal energy etnt and Cauchy stresses
• momentum equation
• internal energy equation
• constitutive law
In the constitutive relations, Cijki is the material response tensor which relates any frame-invariant rate of the Cauchy stress to the velocity strain tensor DIJ ; the tensor Sijki acting on the spin tensor uiki assures the objectivity of the Cauchy stress tensor. It should be noted that pressure gradients describing the propagation phenomenon (acoustic waves) and evaluated usually in fluid dynamics algorithms by approximate Riemann solver together with pure convection fluxes are included here to the Lagrangian phase. The reason for it is that in non-linear solid materials, the propagation phenomenon is more complex then in Newtonian fluids because of coupling of constitutive equations. Thus, it is problematic to construct a very general algorithm based on the approximate Riemann solver capable to deal with any non-linear solid material. A finite element formalism is then used. The computational domain Q C JR3 is approximated by a discretisation Ph using tetrahedrons or brick elements Pe £ Ph • The following discrete spaces are also introduced :
where PI is the space of polynomials in three variables and of degree 1 ; Vh is the approximated value of the velocity on the discretization Ph '•> u/, is the m-dimensional vector containing approximate values of p, eint, stresses and internal variables.
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2.2 Finite Volume formulation for the transport phase
After splitting, the sub-system of the transport phase can be written in the following conservative vector form :
where V 6 Rmx]Q,tfinal] -> ]Rm,F(V} e JRm ->> Rm. The spatial operator of the transport phase sub-system is non-symmetrical. Therefore, an upwind-like procedure must be applied . Dealing with unstructured grids, we use the finite volume method because of its ability to solve hyperbolic problems in a conservative form using flux computations at the cells boundaries. The direct application of a finite volume technique with only one kind of cells (like cell or node centred cells) is not optimal for our ALE algorithm. On one hand, the difficulty lies in the way the physical variables are represented. In the Lagrangian phase some of them are represented with a constant approximation per element like stresses whereas others have a linear approximation such as velocities and displacements. Therefore, a complete use of a cell-centred technique in the transport phase requires a double averaging operation to pass from the nodal representation to the constant per element one and vice-versa, which gives too much diffusion. On the other hand, the necessity to govern the motion of the deformable computational grid does not enable us to use the same spatial location for velocities and element-centred variables, like used in conventional Computational Fluid Dynamics (CFD) on fixed grids. It is due to the fact that the ALE grid rezoning techniques usually deal with nodes attached quantities (for instance, when solving the Laplace equation to regularise the ALE grid). In order to avoid such difficulties, we use two kinds of cells (cell and node centred cells) in a combined finite volume algorithm of the transport phase. The momentum is localised on node-centred cells built around finite element nodes of the discretisation Ph, whereas the other conservative quantities are integrated using cells coinciding with the finite element mesh Pe. To approximate the momentum, the third discrete space is introduced:
where C/ are node centred cells forming an other covering of the domain f2. A variational formulation of the transport phase can be written in general form as follows : Find Vh that satisfies :
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Choosing W'/, in an appropriate way in the spaces Uh or W/j and using Green's formula, we can rewrite (2) on a local level as follows : • momentum equation
with Vi = ( pvi ) , Fik — ( pvi(vk-vk)
), on the node-centred covering and
• other quantities
with
integrated using cell-centred cells coinciding with the finite element mesh Pe. Such integral forms present an advantage to be compatible with the spatial approximation adopted in the Lagrangian phase. Moreover, they allow to deal with moving and deformable grids. To evaluate the convective flux integrals at the interfaces between neighbour polygonal cells, we suit the standard procedures described by many authors (see for example [NKO 94]). However, due to the original split technique presented above, the construction of numerical convective fluxes is simpler than in standard CFD calculations. Indeed, the solution of the Riemann problem is quite simple because the transport phase system is an hyperbolic degenerated system. Thus the Roe scheme based on characteristic variables cannot be applied. Since the upwind direction is uniquely determined by the quantity (v-v).n denoting the scalar product between a convective velocity across the cell boundary and an outward normal to this boundary, we can use the well known full upwind scheme which respects the flow direction. 3. Numerical examples The efficiency of the mixed convection technique presented above has been evaluated on two solid dynamics problems dealing with elasto-plastic material. 3.1 Stress wave propagation problem
The first example is a ID elastic wave propagation problem first considered in [LIU 86]. A stress wave is generated in a long rod by a pressure pulse applied at one end starting att = 0. The rod is discretised in 400 constant-width elements over a length sufficient to avoid wave interactions with the unloaded end of the grid during the whole simulation. The ALE grid is kept fixed until t = 24 sec, then suddenly moved with a positive constant velocity until the final time t — 32 sec. The definition of the problem is following :
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To provide a severe test of the proposed transport algorithm, the grid velocity is taken equal to as much as one fourth of the sound speed (c ~ 1), whereas the material velocity (of the order of 0.01) is almost negligible compared with assumed grid velocity (v = 0.25). The analytical and numerical solutions of this problem are superimposed in the Figure 1. Only the spatial interval 24 < x < 34 is shown in order to closely examine the wave fronts. As may be observed, the implemented transport technique captures well the discontinuities of the analytical solution. Performing correctly the convection of all main variables, the algorithm introduces no high frequency damping because no average operation is used. That provides the good conservation properties of the presented algorithm. We can also note that the algorithm is almost insensitive to the grid motion because very close results are obtained with the fixed grid. To reduce numerical oscillations due to the centred representation of the propagation phenomenon in the Lagrangian phase the artificial damping is used.
Figure 1. Liu's stress wave propagation test, axx profile 3.2 Taylor bar impact problem To validate our mixed FE-FV algorithm in multidimensional case we consider a non-linear solid mechanics problem of a cylinder impact against a rigid wall known as Taylor bar impact problem. This test is typically used to validate fast-transient dy-
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namics computer codes because the final length and shape of the cylinder are very sensitive to the constitutive behaviour of its material. The geometrical and material data of the problem are shown in the Figure 2. The cylinder impacts a rigid, frictionless wall with an initial velocity of 227 m/s. The material is supposed elasto-plastic with isotropic hardening. The final time of the simulation is 80/^s corresponding to a moment when all kinetic energy of cylinder is consumed by the plastic deformation process. As the original problem is axisymmetric we discretise only a quarter of the
Figure 2. Taylor cylinder bar impact cylinder using 1050 hexahedrons (with 50 elements in the length). Because of a large elasto-plastic material flow this test is particularly severe for the purely Lagrangian codes. Due to a fast impact velocity, the computational grid, when attached to the material particles, undergoes so strong local compression that element volumes become negative and calculations numerically "explode". In order to achieve the Lagrangian calculation we use a coarse grid near the impact zone. However, when we use the ALE formulation with a regularisation condition, the computational grid is automatically maintained uniform during all calculation. The Figure 3 shows the evolution of the computational grid in Lagrangian and ALE cases. In spite of a great difference between the grid and material velocities in ALE case, involving a high non uniform convective velocity, the algorithm manages successfully with the transport phase. The values of the final height HF and radius Rf of the cylinder obtained by our mixed transport technique are in good agreement with the results computed by various 2D and 3D codes, as can be seen in the following table. Author [HAL 86] [LIU 86] [CAS 95] present paper
Code DYNA-3D PLEXIS-3C DYRAC++
Formulation ALE ALE Lagr. ALE
Solution 3D axisym. axisym. 3D 3D
Hf (mm) 21.47 21.53 21.47 21.45 21.43
Rf (mm) 7.03 6.87 7.14 7.09 7.11
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Figure 3. Comparison of the Lagrangien (left) and ALE (right) calculations 4. Conclusion A mixed ALE algorithm using Finite Element and Finite Volume techniques has been presented and tested on solid mechanics problems with non-linear material behaviour. One can note that this algorithm initially proposed for simulation of hydrodynamic problems dealing with compressible fluids is able to treat fast solid dynamic problems. Capable of dealing separately with compressible fluids and nonlinear solids, the presented approach will be applied to the numerical study of strong fluid-structure interaction problems. Bibliography [HUE 95] A.HUERTA, F.CASADEI and J.DONEA, ALE Stress Update in Transient Plasticity Problems, 4th International Conference on Computational Plasticity - Fun damentals and Applications, Barcelona, 3-6 April, 1995. [POT 97] S.POTAPOV, A Fast Dynamics ALE algorithm based on a mixed Finite Element Finite Volume approach, Ph.D. Thesis, Ecole Centrale Paris, 1997. [NKO 94] B.NKONGA and H.GUILLARD, Godunov type method on non-structured meshes for three-dimensional moving boundary problems, Comp. Meths. Appl. Mech. Engrg. 113, 1994, p. 183-204. [HAL 86] J.O.HALLQUIST and D.J.BENSON, DYNA-3D : User's Manual, Univ. Of California, Lawrence Livermore National Laboratory, Report UCID-19592, 1986. [LIU 86] W.K.Liu, T.BELYTSCHKO and H.CHANG, An Arbitrary Lagrangian Eulerian Finite Element Method for Path-Dependent Materials, Comp. Meths. Appl. Mech. Engrg. 58, 1986, p.227-246. [CAS 95] F.CASADEI, J.DONEA and A.HUERTA, Arbitrary Lagrangian Eulerian Finite Elements in Non-Linear Fast Transient Continuum Mechanics, JRC, European commission, Report EUR 16327 EN.
An Euler Code that can compute Potential Flow
Mani Rad and Philip Roe Department of Aerospace Engineering The University of Michigan Ann Arbor, Michigan 48109-2140 U.S.A. Centre pour Mathematiques et leurs Applications Ecole Normale Superieure de Cachan, France
ABSTRACT A new approach is taken to the computation of the elliptic part of the Euler equations. In each cell of an unstructured triangular grid, on which the solution is stored at the vertices, the residual is decomposed into purely elliptic and purely hyperbolic contributions. The elliptic part is minimised in a norm suggested by an earlier analysis of linearized potential flow. The minimization is carried out by projecting the direction of steepest descent into a surface on which both the entropy and enthalpy are constant. When the method is applied to subcritical flows for which potential solutions should be obtained, the enthalpy is constant to machine zero, and the entropy is constant to an extremely high degree of accuracy. Key Words: Euler equations, potential flow, fluctuation splitting.
1.
Introduction
This paper contributes further to the development of computational methods for the Euler equations (and eventually for other problems that share their structure) that reflect genuinely multidimensional physics. These methods aim to retain the benefits of a physical aproach to strongly discontinuous flows while avoiding the defects of upwind methods applied to almost incompressible flow. There has recently been considerable success in extending compressible flow
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codes to the low Mach number limit by means of matrix preconditioning. The present paper takes the approach of equation decomposition, which is a very closely related concept. We continue with the analysis in [ROE 99-1] where we showed that the decomposition of the residual into elliptic and hyperbolic parts, and the update procedure are sufficiently simple that explicit formulae can be given for them. This enables the singularities of the procedure, near sonic or stagnation points, to be displayed and percieved harmless. In that paper the update for the elliptic part was performed by a straightforward method of steepest descent. Here we constrain the descent direction so that the elliptic part of the update does not affect the entropy or the enthalpy variables. In this way our Euler code is able to compute with very high precision potential flows for which those variables should be exactly constant.
2.
Forms of the Euler Equations
Different aspects of the Euler equations are most readily expressed by choosing different sets of unknowns. For computing compressible flow, the most fundamental choice is probably the set of conserved quantities
since a weak solution of this form of the equations captures shocks that satisfy the Rankine-Hugoniot conditions. Associated with these variables is the flux tensor There is computational convenience in the parameter vector
where h is the total specific enthalpy (E -\- p ) / p , on account of the property that all components of u, F_ are simply bilinear in terms of z. This allows the construction of local linearizations having conservation properties, both in the one-dimensional [ROE 81] and multi-dimensional [DEC 93] cases. Also important, at least to the present approach, are what have been called [HAY 63] the natural variables
where S is entropy, since it is uniquely in these variables that the Euler equations can be decoupled into maximally independent subsystems [HAY 63][ROE 96-1][ROE 96-3]. This enables the essentially different hyperbolic and elliptic behaviour to be computed independently with the minimum of "crosstalk". In this paper we are particularly concerned with the solution of the elliptic subsystem, which contains only the derivatives of pressure and flow angle.
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Fluctuation Splitting
In this approach [ROE 94] [DEC 95], the computational domain is divided into elements in an unstructured way, with the unknowns stored at its vertices. Iteration toward a steady solution from an in intial guess takes place by computing an average residual for each cell (the fluctuation 4>T in element T), and then changing the current solution at each node of the cell T by an amount proportional to c^T;
where a; is a relaxation factor and the weight aj is a matrix to be determined. Our aim is to create a method of this kind that exploits the different properties of the different sets of unknowns. We represent the solution in terms of the parameter vector, and find the conservative residual by integrating over the cell (of area AT)
where the matrices C •= <9F_ jdz and C — <9F_ fdz are locally constant because of the quadratic property [DEC 93]. Our aim is then to reduce this residual in some suitable norm. The norm is chosen by splitting the residual into its elliptic and hyperbolic parts, which is uniquely possible in the natural variables [HAY 63] [ROE 96-1]
where s,n are streamline and normal coordinates. The first two equations represent the elliptic part and involve only the two unknowns of pressure and flow direction; the second pair are the hyperbolic part. In [Roe 99] the elliptic part was solved by minimising that component of the residual, whereas the hyperbolic part was solved as a pair of scalar advection equations. In this paper the elliptic part of the update is constrained not to interfere with the hyperbolic part and vice versa.
4.
The Update Matrix
This gives a convenient way to think about Fluctuation-Splitting schemes. Let the initial state of a cell (in two dimensions) whose vertices are (a, b, c) be
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represented by the twelve scalar quantities
where w is whatever quantity we have decided to store at the vertices. The update can be represented as a matrix multiplication
where U is some matrix that is constant within the cell and u; is a relaxation constant. For example, suppose we intend to update the solution by a steepest-descent minimisation of
where Q is some symmetric matrix representing a local norm. If we write Q = ptp this is equivalent to an LI minimization of P4>T', which is some weighted combination of the residuals. We will treat <j)T as a linear function RTWT of the vertex values because it arises from a linear process to find the local derivatives, followed by multiplication by matrices that are frozen during the update. In general R is a rectangular matrix formed from three 4 x 4 blocks (see the examples below). Hence the quantity to be minimized is
The gradient of this is
and so the update procedure within each triangle
where W is some global constant, implements the required procedure. If we had chosen to store the natural variables x it is easy to show that the four residuals 0X in (7) are given by
where, using standard methods to evaluate the derivatives,
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with u nj = [u(Aj,)j - v(Ax)j] and u sj = [t>(A y )j + u(&x)j] with ( ( & x ) j , (A y )j) the vector representing the side opposite vertex j. Thus s,n now represent directions along and normal to an edge. The first two components of this vector comprise the elliptic part of the problem, if M < 1, so the quantity to be minimized is where where the relative weighting of the continuity and vorticity residuals follows [ROE 96-31. Then the update matrix is
Since in fact we are storing the parameter vector z we must estimate the quantity to be minimised as
and the update becomes
This was the process studied in [ROE 99-1], merely as a first step. It has the disadvantage that minimising the elliptic part of the residuals changes the convected quantities, entropy and enthalpy. We replace it here by a constrained minimization, in which the elliptic part of the residual is reduced as quickly as possible subject to the constraint that this part of the procedure should not change the convected quantities.
5.
Constrained Update Procedure Following [ROE 99] the unconstrained update matrix can be written
where from now on the matrices R incorporate the change of variable dx/dz For example Ra is the matrix
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where
and
After exploiting some coincidences each block of the update matrix becomes
Since we intend complete decoupling at the update level between the elliptic and hyperbolic parts, the elliptic part of the residual should only contribute to changes in pressure and flow angle and not to entropy and enthalpy. Conversely the entropy and enthalpy residuals should change only the entropy and enthalpy. For each natural variable, the direction of a vector along which only that variable changes is
An arbitrary change in the parameter vector dzu coming from the unconstrained minimisation can be expressed in this basis by solving dzu = Pa where a = (as, c*/,, ap, QeY is a vector of coefficients and P = [rs r^ rp TQ\ is the direction matrix. The constrained changes due to the elliptic part are then found by supressing as,h- This is to say,
where the overbar indicates taking the last two components. The constrained update matrix is therefore
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and carrying out the matrix multiplications reveals
The unconstrained steepest descent has the property that changes due to any triangle sum to zero, thereby ensuring conservation. To preserve this property it is necessary to project each triangle residual onto some local set of directions P and then use these for the changes induced by that triangle at each of its nodes. Close to M = 0 these directions degenerate; rs, r^ and rp become nearly parallel. Under these circumstances we found that the code would not converge. To achieve convergence we needed to use a nonconservative version in which the changes at each node were projected onto the allowable directions. However, at higher Mach numbers the conservative technique worked well. 6.
Computational Examples
The first figure presented is a potential flow calculation on a symmetrical airfoil at zero incidence, where the least-squares method is used only in the subsonic region. Where a supercritical region exists, the hyperbolic part of the problem is handled using the PSI advection scheme. No special procedure was used to match the calculations across the sonic line or across the shock. The last two figures show Euler computations on a cylinder in a subcritical case (Moo = 0.35) and in the incompressible limit (Moo = 0.01) using the residual decomposition scheme described in this paper. The elliptic part of the problem was handled using the constrained least squares scheme as presented in sections 3,4 and 5 while the LDA upwind advection scheme was used on the hyperbolic part. Results shown are for a grid consisting of about 3544 cells and it can be seen that a very small amount of spurious entropy is generated very close to the cylinder wall and especially in the stagnation regions. The rate of entropy generation is measured by
and is measured to be 1.0 x 10 9 for the incompressible case and 1.0x10 ' for the subcritical case. Not shown in the figures but of certain importance is that
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enthalpy stays everywhere constant and equal to its initial farfield value. These results reveals that the projection ideas discussed in section 5 are completely successful for the enthalpy and satisfying for the entropy. In any case, it seems that the present formulation provides a solid framework for treating sensitive regions around the wall boundary. The accuracy of the scheme was measured by comparing the numerical solution to the analytical solution in the incompressible case. Error is defined as (v is velocity vector)
and was measured to be 9.0 x 10-4 for a coarse grid of 860 cells and 2.3 x 10 -4 for a finer grid of 3544 cells. In other words, the scheme's convergence behavior is very close to second order.
7.
Concluding Remarks
The present contribution is still not yet intended to provide a practical recipe for solving the Euler equations. The main defect is that steepest descent is, by itself and even with the improvements derived here, a very slow way to solve nonlinear equations. And applied to a linear elliptic system it amounts only to a point Jacobi relaxation. However, it forms the starting point for many other (mainly Newton-like) methods that are extremely efficient, and which have the same fixed points. However, the objective here is entirely to find an iterate that has a good fixed point, not one that reaches some fixed point quickly. It also remains to be proved that an analogous method will work in three dimensions. There is no problem formally. All the formulae given here extend straightforwardly. The issue is whether the streamwise vorticity that becomes coupled to the potential flow in three dimensions [ROE 96-2] will be well captured. It is significant that discretisations exist [ROE 99-2] for which correct growth of the discrete vorticity is automatic, in the sense that a discrete Kelvin Theorem is guaranteed. Future work will determine whether this is possible.
8.
References
[DEC 93] DECONINCK, H., ROE, P. L., STRUUS, R. J., <>, Computers and Fluids, 22, p215, 1993. [DEC 95] DECONINCK, H, PAILLERRE, H., ROE, P. L.,
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[HAY 63] HAYES, W, PROBSTEIN, R. F., Hypersonic Flow Theory, Academic Press, 1963. [ROE 81] ROE, P. L., « Approximate Riemann solvers, parameter vectors and difference schemes », J. Comput. Phys., 43, pp357-372, 1981. [ROE 94] ROE, P. L., « Multidimensional upwinding, motivation and concepts », von Karman Institute Lecture Series 1994-04 [ROE 96-1] ROE, P. L., MESAROS, L. M., ^Solving steady mixed conservation laws by elliptic/hyperbolic splitting », 13th International Conference on Numerical Methods in Fluid Dynamics, Monterey, July, 1996. [ROE 96-2] ROE, P. L., TuRKEL E.,
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Figure 1. Subsonic (M Mach contours.
= 0.36) and transonic (M
= 0.85) NACA 0012,
Figure 2. Cylinder flow in the incompressible limit (M entropy contours. Euler solution.
= 0.01), pressure and
Figure 3. Cylinder flow in the subcritical range (M entropy contours. Euler solution.
— 0.35), Mach and
Finite volume evolution Galerkin methods for multidimensional hyperbolic problems M. Lukacova-MedvicFova 13 , K. W. Morton 2 , G. Warnecke1
l
lnstitut filr Analysis und Numerik, Otto-von-Guericke-Universitdt Magdeburg, PSF 4120, 39 106 Magdeburg, Germany, 2 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom (also Oxford University Computing Laboratory), 3 Institute of Mathematics, Faculty of Mechanical Engineering, Technical University Brno, Technickd 2, 61639 Brno, Czech Republic
ABSTRACT The finite volume evolution Galerkin method couples a finite volume formulation with an approximate evolution Galerkin operator, which takes into account all of the infinitely many directions of propagation of bicharacteristics for multidimensional systems. Piecewise linear recovery yields second order accuracy even with the first order approximate evolution operator. Numerical comparisons of the evolution Galerkin schemes with the commonly used finite volume methods for the wave equation system and for the Euler equations are presented. Key Words: genuinely multidimensional schemes, hyperbolic systems, wave equation, Euler equations, evolution Galerkin schemes, finite volume methods
1. Introduction
It is our belief that the most satisfying methods for approximating evolutionary PDE's are based on approximating the corresponding evolutionary operator. In order to construct a genuinely multidimensional numerical scheme for hyperbolic conservation laws all of the infinitely many directions of propagation of bicharacteristics have to be taken into account.
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This is the main idea of the evolution Galerkin (EG) methods, which evolve the initial data using the bicharacteristic cone and then project them onto a finite element space. In the recent paper Lukacova, Morton and Warnecke [LMW 99] three new first order evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system are derived and analysed. It is shown that the evolution Galerkin scheme (denoted in the paper [LMW 99] as the EG3 method), which is based on the general theory of bicharacteristics for hyperbolic conservation laws gives the most accurate numerical scheme. This is a base for the second order scheme, which will be constructed using a piecewise linear recovery. In the recent years the most commonly used numerical shemes for hyperbolic conservation laws were the finite volume methods, which are based on a type of directional splitting and on an approximate solution of one-dimensional Riemann problems. Their popularity is particularly due to the simplicity of their formulation as well as implementation. However, it is a known fact that these methods can produce structural deficiences in the solution for some special multidimensional problems (see, e.g., [FEY 92], [LEV 97], [LMW 98], [LMW 99]). The finite volume evolution Galerkin method combines advantages of both approaches: the simplicity of the finite volume formulation and the multidimensionality of the evolution Galerkin schemes. 2. Finite volume evolution Galerkin methods
Consider a general hyperbolic system in d space dimensions
where l , . . . , d represent given flux functions and the unknown functions are U = . Let us denote by E ( s ) : (Hk the exact evolution operator associated with a time step s for the system (1), i.e.
We suppose that Sph is a finite element space consisting of piecewise polynomials of order p 0. Let be an approximation in the space S^ to the exact solution at a time tn > 0 and take Er : Srh -> (Hk(IRd))m to be a suitable approximation to the exact evolution operator E(r), r 0. We denote by Rh : S a reconstruction operator, r > p > 0 . In the present paper we
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shall limit our consideration to cases of constant time step A/, i.e. tn = r?A/, and of a uniform mesh consisting of d-dimensional cubes with a uniform mesh size h. Definition 1. Starting from some initial value U0_ at time t= 0, the finite volume evolution Galerkin method (FVEG) is recursively defined by means of
where the central difference v(x -f h/2) — v(x — h/2) is denoted by STv(x) and represents an approximation to the edge flux difference. The cell boundary value is evolved using the approximate evolution operator ET to tn + T and averaged over 0 < r < At and along the cell boundary, i.e.
where - is the characteristic function of . There are several advantages to this formulation. The most important is that the first order accurate approximation E T to the evolution operator E ( r ) yields an overall second order update from U_n to U_n+ l. To obtain this second order approximation in the discrete scheme it is only necessary to carry out a recovery stage at each time level to generate a piecewise linear approximation U = RhU_n from the piecewise constant to feed into the calculation of the fluxes. In the next section we illustrate this procedure for the wave equation system in two space dimensions. 3. Wave equation system
The wave equation can be written down as a first order hyperbolic system
with the unknown functions . Consider a cone with the apex P — (x, y, t-\At) and the base points Q = Q(0) = (x+cAt cos 0, y+cAt sin 0, t) parametrized by the angle 6 £ [0, 2]. Denote by P' = (x, y, t) the center of the base of the cone. The lines from Q(0] to P generating the mantle of the so-called bicharacteristic cone are called bicharacteristics, see, e.g., [LMW 99] for more details. Using the theory of bicharacteristics it can be shown that the solution (, w, v) at the point P is determined by its values on the base as well as on the mantle of the characteristic cone and the exact evolution formulae can be derived. In the
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recent papers Lukacova, Morton and Warnecke [LMW 98], [LMW 99] several approximate evolution operator for the wave equation system were analysed. It was shown that the following approximate evolution operator leads to the best first order scheme in terms of accuracy, see [LMW 98], [LMW 99]. 3.1. First order approximate evolution operator
Denote by Ph L2 - projection onto a space of piecewise constant functions in IR2, then we obtain the first order scheme which is in [LMW 99] referred to as the EG3 scheme. Space integrals coming from the projection step are computed exactly, i.e. no numerical quadrature is used. The finite difference formulation can be found in [LMW 99], where the coefficients of the scheme are given explicitely. 3.2. Second order reconstruction In order to construct the second order FVEG scheme we take the first order accurate approximate evolution operator (6) - (8) and define a bilinear reconstruction Rh- There are many possible recovery schemes, which could be used. For our computation we choose a discontinous bilinear recovery using a four point averages at each vertex, but others can be used as well. It is taken to be conservative and given as
where tation is used for averaged value
an analogous nooy. For the computation of fluxes through cell edges the time has to be known, see (4). Instead of exact time integration
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the second order midpoint rule is used, i.e.
Two dimensional space integrals of bilinear functions Rh with respect to 9 and S which occur in (9) are computed exactly without any numerical quadrature and thus all of the infinitely many directions of propagation of flow information are taken explicitely into account. The above construction leads to the overall second order scheme, which gives in regions of smooth solution very accurate results even on coarse grids, see Table 1 below. 3.3. Numerical results We consider the space periodic problem for the wave equation system (5) with the initial data
In this case the exact solution can be easily computed [LMW 99]. In Table 1 we compare errors of several second order schemes, namely the second order FVEG method, the Lax-Wendroff method (LW) and the standart finite volume fluxvector directional splitting method (FV-FVS), which uses a MUSCL technique for the flux computation and a second order Runge-Kutta time approximation. For the details on the latter one see, e.g., [KRO 97]. We use meshes of 20 x 20, 40 x 4 0 , . . . , 640 x 640 cells and compute also the experimental order of convergence (EOC) from two meshes of sizes NI and N2 as
In all cases the results are for a CFL-number v of 0.45 and an end time of T = 0.2. Experiments for several other values of v and T confirm the second order accuracy of the FVEG scheme. N / \\LL(T)-Un\\
20 40 80 160 320 640 EOC
FVEG 0.008647 0.001789 0.000925 0.000290 0.000080 0.000021 1.94
LW 0.065500 0.016472 0.004133 0.001033 0.000258 0.000064 1.99
FV-FVS 0.057748 0.014502 0.003617 0.000905 0.000227 0.000057 2.00
Table 1. Comparison of accuracy of the FVEG method, the Lax-Wendroff method and the FV flux-vector splitting method
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4. Euler equations
The system of Euler equations describing the motion of compressible flow in two space dimensions can be written in the form of hyperbolic system (1) with d — 2; the definition of U_ and Fi(LO is well-known and can be found e.g. in [KRO 97]. If R = (rj n, r 2 i n , £3n, r_4n) denotes the matrix of right eigenvectors we can decompose the vector of conservative variables U_ for any direction n in the following way
A general procedure for derivation of the exact evolution operator for hyperbolic problems can also be applied to the Euler equations. For linear problems this procedure works with the characteristic variables W = R~ U_. Now, instead of W the vector a is used. The following approximate evolution operator for the Euler equation was derived in [OST 97]
where O denotes the unit sphere, is a local speed of sound and u represents the fluid velocity. After L2 - projection onto a space of piecewise constants we obtain the EG scheme for the Euler equations. It was shown by Ostkamp [OST 97] that Fey's method of transport [FEY 92] for the Euler equations can be reinterpreted as the above evolution Galerkin scheme. 4.1 Numerical results We take the well-known test problem, namely the two-dimensional Sod's problem and compare the behaviour of the evolution Galerkin method (11) and the LeVeque wave propagation algorithm [LEV 97], which is available as a public domain software package called CLAWPACK. In order to avoid a discussion of the limiters we compare here only first order schemes. Note however that in LeVeque's wave propagation algorithm also first order correction terms for xy-cross derivative are included. The computational domain is the square [-1,1] x [-1,1]. To ensure the CFL stability condition, the CFL number is taken 0.8. We choose periodic boundary conditions and the following initial data
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Solutions at time T = 0.2 are computed on a quadrilateral grid with 200 x 200 grid cells. In Figures 1 and 2 the isolines of solution obtained by the evolution Galerkin method (11) and by the wave propagation algorithm, respectively are drawn. The {/-velocity is not depicted since it is symmetric to the x-velocity. For the evolution Galerkin scheme the resolution of the flow phenomena is the same in all directions and information is moving in infinitely many directions in a circular manner. However, we can notice that the wave propagation algorithm, which makes use of an improved directional splitting, does not preserve circular symmetry in such a good manner as our scheme and some dependence of the solution on the grid can well be seen.
Figure 1. Evolution Galerkin scheme: density (left), x-velocity (middle), pressure (right)
Figure 2. Wave propagation algorithm: density (left), x-velocity (middle), pressure (right)
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5. Acknowledgement
This research has been supported under the DFG Grant No. Wa 633/6-1 of Deutsche Forschungsgemeinschaft, and partially by the Grant No. 201/97/0153 of the Czech Grant Agency and by the DAAD. 6. Bibliography
[FEY 92]
FEY M., JELTSCH R., «A simple multidimensional Euler schemes, Proceedings of EC COM A S'92, Elsevier Science Publishers, Amsterdam, 1992, p. 667-671.
[KRO 97]
KRONER D., Numerical Schemes for Conservation Laws, Wiley - Teubner, Stuttgart, 1997.
[LEV 97]
LEVEQUE R.J., «Wave propagation algorithms for multidimensional hyperbolic systems», J. Comp.Phys., 131, 1997, p. 327-353.
[LMW 98]
LUKACOVA - MEDVIDOVA M., MORTON K.W., WARNECKE G., «On the evolution Galerkin method for solving multidimensional hyperbolic systems», Proceedings of ENUMATH'98, World Scientific Publishing Company, Singapore, 1998, p. 445-452.
[LMW 99]
LUKACOVA - MEDVIDOVA M., MORTON K.W., WARNECKE G., Evolution Galerkin methods for hyperbolic systems in two space dimensions^, Preprint University of Bath, England, 1999, submitted to MathComp.
[OST 97]
OSTKAMP S., «Multidimensional characterisitic Galerkin schemes and evolution operators for hyperbolic systems, Math. Meth. Appl Sci., 20, 1997, p. 1111-1125.
Nonlinear anisotropic artificial dissipation Characteristic filters for computation of the Euler equations Thorsten Grahs, Andreas Meister and Thomas Sonar Institut fur Angewandte Mathematik Universitdt Hamburg, Bundesstr. 55 20146 Hamburg, Germany ABSTRACT We employ a nonlinear anisotropic diffusion operator like the ones used as a means of filtering and edge enhancement in image processing, in numerical methods for conservation laws. It turns out that algorithms currently used in image processing are very well suited for the design of nonlinear higher-order dissipative terms. In particular we stabalize a central scheme, known for its oscillating behaviour by the construction of a nonlinear diffusion term. Using information from the data a so called structure tensor following from ideas of anisotropic diffusion in image processing due to Weickert is constructed, containing information about the orientation and strength of the necessary diffusion. In particular this means constructing a diffusion matrix consisting of eigenvectors parallel and perpendicular to discontinuities and eigenvalues denoting the amount of dissipation depending on the local strength of the gradients. These directions are used to steer the amount of dissipation which means surpressing diffusion across the shock front and using the perpendicular direction to enable the necessary diffusion to stabilize the underlying second order scheme. The diffusion terms are based on the artificial compression method (ACM) due to Harten and the extensions of Yee. The limiting takes place on characteristic based diffusion terms acting on the characteristic velocity. Numerical results are shown for the two-dimensional Euler-equations. Keywords: low dissipation, central finite difference schemes, shock-capturing methods, anisotropic diffusion, characteristic filters, Euler equations
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1. Introduction
The construction of suitable artificial viscosity terms for stabilizing finite volume and finite difference schemes of higher order is a difficult task. In the last decade we observed therfore a strong tendency to construct numerical approximations of conservation laws without explicit knowledge of their numerical diffusion. The modern total variation diminishing (TVD) or essentially non-oscillatory (ENO) shemes belong to this class, in which a basic firstorder scheme is improved by the use of sophisticated recovery functions, see [Son97, MS99]. There, are however, certain circumstances in which an approach using explicit construction of artificial dissipation would be advantageous. If we consider pseudospectral methods the concept of ENO recovery is very hard to apply if the degree of the polynomials used is high. Here one would like to compute shocked solutions with central schemes and post-process the numerical solution such that high frequency oscillations are filtered, and shocks are steepend and represented with high resolution. Over the years there were no general attempts to derive a constructive theory which would enable the design of suitable artificial viscosities within the CFD community. However, filtering and edge enhancement is a fundamental task in image processing and in recent years a theory of nonlinear anisotropic diffusion was created and can now be found in textbooks like [Wei98]. In a noisy picture one also would like to filter the high frequency components before detecting the edges (i.e. jumps in grey level). Then one would like to enhance the edges in order to represent the edges in high resolution. Now there is nothing which keeps us from interpreting our numerical solution corresponding to the conservation law as a photograph or picture. In the same way the photographer would very much prefer to see the contours on his picture as sharp thin lines the numerical analyst would prefer to see shocks as crispy lines instead smeared thick regions. To accomplish this, the picture as well as the numerical solution have to be denoised. After removing the high frequencies we would like to spend a dose of diffusion tangential to shocks - that is what anisotropy is all about in this context. In contrast, in the vicinity of shocks we would like to solve a kind of nonlinear anisotropic backward heat equation to enhance the structure of a shock front. Devices and algorithms satisfying exactly these requirements are ready to use if one is willing to enter the area of image processing. The aim of this paper is mainly to show how one can use ideas from image processing in the area of the numerical treatment of conservation laws. We have used suchs methods succesfully for scalar mixed Burgers-advectionequation namely stabilizing a Lax-Wendroff scheme and surpressing the oscillatory behaviour of the scheme in the vicinity of shocks (see [GMS98]). Here we would like to discuss the extension to the Euler equations. First we describe the governing equations and the basic scheme. In the following we construct the so called structure tensor to gain informations about the local strength and orientation of the shock. This information will be used to develop a diffusion
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matrix to steer the amount of diffusion necessary to stabilize a second order central scheme.
2. Anisotropic diffusion applied to characteristic filters 2.1.
Governing Equations
We consider the two dimensional time-dependent Euler equations for a compressible gas in equilibrium in conservation form, i.e.
in which
denote the vector of conserved quantities and the flux functions in x and y, respectively. If t denotes time and x = ( x 1 , x t } T 6 space coordinates, the mapping
denote density, velocity, pressure, total energy and enthalphy, respectively. Enthalphy is defined by
To close the system, an equation of state is needed. For ideal gases one uses
where 7 denotes the ratio of specific heats. In the case of dry air one assumes a value of 7 = 1.4.
2.2.
Construction of the characteristic filter
We discretize on a cartesian grid with mesh size Ax, Ay, respectively. Since in this case finite volume and finite difference techniques are equivalent, we use in the following a typical finite difference notation. We start from an explicit central scheme written as
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where the numerical flux functions include the filter terms and hence the cross diffusion parts and can be written as follows:
The filter operators LA, LB depend on the Jacobian-matrix A and B of the flux function F and G, respectively. The filter terms Lc models some kind of cross diffusion and depends on a combination of the Jacobians, namely
(see [Hir90] for details). The vector n : = (nx'1,nxl)T is chosen in direction of the cell diagonals, i.e n+ := (l,l) T /\/2 and n~ :~ ( 1 , — l) T /\/2- The matrices are all evaluated as a Roe average ( see [RoeSl]). So one writes the corresponding filter terms as
and similar for the other terms. Here R^'> denotes the matrix of the right eigenvectors corresponding to the appropriate matrices written as superscript. <$(') denotes the real filter term. In the basic construction, we follow the recent paper of Yee and her co-workers [YSD99], based on the Artificial Compression Method (ACM) of Harten and the extensions of Yee (see f.e.[Har78, Har83, Yee85]). We extend these method by the integration of a directional based - that its what we denoted with anisotropic - diffusion known from image processing. So the elements of 3>('^ in (1) are denoted by ^'\l — 1 , 2 , 3 , 4 . We give here as an example the construction of the term <j>i+\/2 •• All other terms are constructed in the natural extension of this definition depending on the matrices and grid points were they are evaluated. So the elements of j 1/2 • write as
The denotes the weighting coefficients steering the anisotropic diffusion. The construction of this coefficient will be described in the following section. The
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choice of the is exactly the same like the one described in [YSD99] with the use of the limiter functions gl- as
Additional possible choices for this function and a detailt description are given in [YSD99]. The definition of will be give below. The choice of the filter function depends on the characteristic splitting of as the flux representation of the gradients. Thus, we consider elements The different a depend again on different matrices of the right eigenvectors depending on the gridpoint were they are evaluated. They are given as:
(2)
Hence oli+1/21j+1/2corresponds to a1i+1/22j+ 1/2 and so on.... The filter function writes as (3)
(4)
The a' + 1 / 2 in (3),(4) are the characteristic speeds of the corresponding Jacobians, which are equal to their eigenvalues. This term can also be weighted with the diffusion coefficient /3li+1,2 . steering the amount of dissipation corresponding to this direction.
2.3.
The structure tensor
We start from the so called structure tensor
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smoothed version of the characteristic gradients R 1 AC/,5, where U$ is a presmoothed version of the data Un. This means convolution with a Gaussian kernel whith convolution scale 6, i.e
Since in the continnous case this is equivalent with solving the heat equation we apply this to the data Uij with stopping time T = ^S2, where S is a parameter which has to be chosen. In order to remove the small scale oscillations by means of this smoothing technique we define S depending on the grid size h := ^/NxKy. Consequently, the structure tensor reads as (5)
with
where a1''5 corresponds to the smoothed data U$. Therbye one can also use a smoothed version of the structure tensor (5), i.e (6)
which means component-wise convolution with scale v, which denotes the width of the averaging region. In practice we are solving the heat equation for each component seperately.
2.4.
The diffusion
matrix
After having constructed the structure tensor we are going to compute the eigenvectors and eigenvalues of (6), (7)
(8)
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Therbye the corresponding eigenvalues are given by (9)
Now we construct the dissipation matrix D by introducing the ansatz due to Weickert [Wei98]:
(10) Here Vv denotes the matrix of the eigenvectors (7), (8) and A,, = diag(/i,/2) represents a diagonal matrix where the diagonal elements have to be calculated in a convenient manner as described below. In order to recover shocks (or, equivalently, in order to enhance edges) the diffusivity l\ perpendicular to edges should be reduced if the contrast \\^ is high. This can be achived by an anisotropic regularization of the Perona-Malik model [PM90] also adapted from Weickert:
The values of m and Cm are chosen in such a way that the so-called flux 4>(s) := s$(s) is increasing in an interval s 6 [0, A] and decreasing in s g]A, oof. The choices depend on a one-dimensional analysis of the Perona-Malik model. In agreement with Weickert we chose m — 4 and thus €4 — 3.31488. The so-called contrast parameter A, separating areas with forward (low contrast) from backward (high constrast) diffusion, can be chosen freely. Based on numerical experiments it turns out that for calculations concerning systems it is usefull to have an adaptive parameter instead of a fixed one. We calculate for each variable the maximum of the according gradients in every timestep and choose a fixed percent of this maximum to determine the parameter A. Thus, the diffusion matrix (10) can be written as
(11) with coefficients
where V\.}P — (VII->P, ^i 2 i p) T - The superscript / reminds that the diffussion matrix (10) and so the structure tensor (5) resp. (6) have to be computed for every characteristic variable seperatly.
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Weickert proved that there is always a finite difference stencil which leads to a stable scheme. Moreover, he was able to show that three directions suffice to guarantee a non-negative discretization, since a negative discretization is equivalent to an ill-posed problem. The discretization of a 3 x 3 stencil reads as follows:
This gives the weighting coefficients for the dissipative fluxes which leads to a steering of the dissipation terms depending on the magnitude of the gradients as described it in the last section.
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3. Numerical results As an example we use a test case given by LeVeque [LeV93] with initial data
and Ui = (2, 0, 0,15) T , Ur = (1,0,0,1) T in primitive variables. The solution consist of a shock running outwards followed by a rarefaction wave and a contact discontinuity. A second shock moves inwards towards the center. We use an equidistant discretization with Ax = Ay = h = 0.025 and a CFL-Number = 0.4. The smoothing parameters are given with 8 = 0.25/J. and v = 0.0 which means no smoothing of the structure tensor takes place. The contrast parameter A is chosen as 0.4max(Ai ; i x ).
Figure 1: Solution of the test case (12) at time t=0.13
4. Concluison We have extended the characteristic filter approach by Yee, Sandham and Djomehri [YSD99] by an anisotropic directional based diffusion from image processing. Since this integration into the field of numerical conservation laws is a novel approach, the calculations presented here are preliminary results and will need further research. Overall we are quite optimistic concerning the behaviour of the suggested extension of this scheme.
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References [GMS98] Th. Grabs, A. Meister, and Th. Sonar. Image Processing for Numerical Approximations of Conservation Laws: Nonlinear anisotropic artificial dissipation. Hamburger Beitrage zur Angewandten Mathematik, Reihe F: Computational Fluid Dynamics and Data Analysis 8, 1998. (submitted to SI AM J. Sci. Comp.). [Har78]
A. Harten. The artificial compression method for computation of shocks and contact discontinuities. III. self-adjusting hybrid schemes. Math. Comp., 32:363-389, 1978.
[Har83]
A. Harten. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 49:357-393, 1983.
[Hir90]
Ch. Hirsch. Numerical computation of internal and external flow, volume 2. J. Wiley fe sons, 1990.
[LeV93] R.J. LeVeque. Simplified multi-dimensional flux limiter methods. In M.J. Baines and K.W. Morton, editors, Numerical Methods for Fluid Dynamics 4, pages 175-190. Oxford University Press, 1993. [MS99]
A. Meister and Th. Sonar. Finite Volume Schemes for compressible fluid flow, in press: Surveys of Mathematics in Industry, 1999.
[PM90]
P. Perona and J. Malik. Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach.Intell, 12:629-639, 1990.
[RoeSl]
P.L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43:357-372, 1981.
[Son97]
Th. Sonar. Mehrdimensionale ENO-Verfahren. Advances in Numerical Mathematics. B.G.Teubner Stuttgart, 1997.
[Wei98]
J. Weickert. Anisotropic Diffusion ner, Stuttgart, 1998.
[Yee85]
H. C. Yee. Construction of explicit and implicit symmetric TVD schemes and their application. J. Comput. Phys., 68:151, 1985.
in Image Processing. B.G. Teub-
[YSD99] H. C. Yee, N. D. Sandham, and M. J. Djomehril. Low-dissipative high-order shock-capturing methods using characteristic-based niters. J. Comput. Phys., 150:199-238, 1999.
Nonlinear projection methods for multi-entropies Navier-Stokes systems
Christophe BERTHON
Frederic COQUEL
ONERA,
LAN-CNRS,
BP 72, 92322 Chdtillon Cedex, FRANCE.
4> place jussieu, 75252 Paris Cedex 05, FRANCE.
ABSTRACT This paper is devoted to the numerical approximation of the compressible Navier-Stokes equations with several independent pressures. Several models derived in plasma physics or in turbulence typically enter the proposed framework. The striking novelty over the usual Navier-Stokes equations stems from the impossibility to recast equivalently the present system in full conservation form. Classical finite volume methods are shown to fail in the capture of shock layers. We propose a new method, the so-called nonlinear projection operator, for correcting the errors while preserving all the stability properties. Key Words: Navier-Stokes equations, Non conservative products, Nonlinear projection.
1. Introduction The present work treats the numerical approximation of the solutions of the Navier-Stokes equations for a compressible fluid modelled by two independent pressures, e.g. each of the pressures comes with its own specific entropy. Despites that such models are seen to exhibit several close relationships with the usual Navier-Stokes system, the fundamental discrepancy stays in the lack of an admissible change of variables that recasts the system in full conservation form. None of the entropy balance equations boils down to a conservation law involving non conservative products that account for dissipative phenomena : namely the entropy dissipation rates. Such systems occur in several distinct physical settings. They arise for instance in plasma physics and they can be also recognized within the frame
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of the "two transport" equations models for turbulent compressible flows. All these models are addressed below with the assumption of a large Reynolds number. The numerical capture of the viscous shock layers is of primary importance in the present work. Since the Reynolds numbers of interest are large, these layers display the character of a shock wave in that they differ from their end states only in a small interval of rapid transition. Our purpose is to correctly capture the two end states, VL and VR, of a given shock layer together with its speed of propagation a whitout resolving sharply the viscous layer itself. It is quite well-known that such an issue does not raise special difficulties within the standard frame of the Navier-Stokes equations, e.g. in conservation form. At the very core of this success, is the definition of v#(cr; v^) stays free from the entropy dissipation rate and in this sense, its numerical capture stays also free from the rate of numerical dissipation. The situation turns out to be completely different in the setting of the extended Navier-Stokes equations. Its non conservation form makes this time the triple (cr; VL, v#) to heavily depend on the precise shape of the diffusive tensor. Such a dependence stays at the basis of recent works devoted to hyperbolic systems involving non conservative products (see Lefloch [6], Dal Maso-LeFlochMurat [2], Raviart-Sainsaulieu [8]). In the setting of numerical methods, this dependence implies that the numerical viscous tensor must fit the exact last one. The negative consequences can be found in the numerical result presented below. In the present work, we propose an approach based on the analysis of the discrete dissipation rates of a given Lax entropy. It turns out that the end states require for their correct capture to prescribe explicitely the Lax entropy dissipation rate. This requirement asks the numerical methods to satisfy in turn an imposed rate of entropy dissipation. This non standard issue is precisely the main motivation of the present work. Let us underline that the results we state below extend in a straightforward way to higher space dimensions, taking advantage of the rotational in variance of the equations. 2. Mathematical model
We consider a gas with density p and velocity u, which is modelled by two independent pressure laws p and pT, associated with two constant adiabatic exponents 7 > 1 and 7r > 1. The system of PDE's that governs such a fluid model writes :
(1)
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where the involved temperatures respectively read T — p/'p and TT — pT/pThis convective-diffusive system can be understood as an extension of the standard Navier-Stokes equations when considering an additional PDE for governing an additional pressure. Depending on the closure relations for the viscosities /i, [ir and the thermal conductivities K and KT, several distinct physical models enter the present framework. In this section, all these transport coefficients are assumed to be fixed positive constants for the sake of simplificity in the discussion. The smooth solutions of system (1) obey additional governing equations as we now state : Lemma 1 Smooth solutions of (1) satisfy the following conservation law: (2)
where the total energy pE is defined by pE — ^^—I—^y H—^~[- Smooth solutions satisfy in addition the following balance equations : (3) (4)
where the specific entropies are respectively given by s Consequently, smooth solutions of (1) obey :
where the right hand side follows under the assumption of two constant viscosities p, and IJLT . The three balance equations (2), (3) and (4) can be proved to be the only non trivial additional equations for smooth solutions. As a consequence the discrepancy stays in the lack of four non trivial conservation laws. Indeed, none of the equations (3), (4) and (5) boils down to a conservation law and (1) cannot be recast in full conservation form. After the works by LeFloch [6] and Raviart-Sainsaulieu [8], the non conservation form met by (1) makes the end states of shock layers to depend on the closure relations for the coefficients //, \JLT and /c, KT. In order to assess this issue, let us focuse on the non standard balance equation (5) where by contrast with (3) and (4) which dissipation rates are independently imposed, (5) exhibits a compared rate of both the entropy dissipations. The idenity (5) continues to play a major role in a general setting since they encode a generalized jump condition which turns out to play a central role for our numerical purpose :
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Theorem 2 Assume that p,, ^T, K and KT denote positive constants. Then the triple (cr, VL, vR) associated with the resulting dissipative tensor necessarily obeys the jump relations :
(6.i) (6-ii) (6.iii) and necessarily satisfies the two entropy inequalities : (7)
Under an appropriate setting (see the companion paper [I]), (7) can be specified as follow :
(8) defining a generalized jump condition where the involved averages find a unique definition (see [1] for details). Remark 3 In view of the jump relations (6) and (8), one of the two entropies, either ps or psT, must be understood as a nonlinear function of the four remaining independent variables (p,pu,pE,.}.
3. Godunov methods with nonlinear projections For the sake of simplicity , we do not address the discretization and we set K = KT — 0 We refer the reader to the companion paper [1] for the required discrete formulae. 3.1. L2 projections (tn -+ tn+1'~) An equivalent system using the conservation laws for p, pu, pE and the evolution law governing psT is considered. Notice that the evolution law for ps can be understood as an additional law satisfied by the smooth solutions. Choosing consitant formulae, based on a Godunov method for instance, the discrete solution satisfies, formally, the following dissipation rates of entropie :
(9)
(10)
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As a consequence of the error which occurs in (10), the expected generalized jump relation (8) is systematicaly violated. The reader is referred to [1] for a rigourous proof and to the numerical results below for an illustration of the negative consequences of such a failure.
3.2. Nonlinear projection methods (tn+1,-- --- tn+1 ) We propose to add an additional step to classical L2 projection methods, the nonlinear projection step, which purpose is to correct the errors. In order to preserve the required conservation properties, let us define :
Setting vf + i = ( / 9^ I '-,(p W ) l n + i '-,( / 9E)^ 1 '-,(p5 r ) t ni + 1 ), to enforce the validity of the generalized jump condition at the discrete level, we propose to seek for (pST)™+11as a solution of : :
The above nonlinear problem in the unknown (psT}™+1 can be shown to admit a unique solution as soon as the approximate Riemann solver involved in the first step obeys discrete entropy inequalities for the Lax pair (ps,psu). The nonlinear projection step (12) allows to prove in addition the following stability results : Theorem 4 Under the required CFL restriction, the following discrete entropy inequalities are satisfied :
for all strictly decreasing and concave functions <j> and if). The following maximum principles for the specific entropies s and ST are met :
Both the pressures Pni+1 and PTin+1 stay positive as soon as the density pn+1 i s positive.
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The proof of the above statement is detailled in [1].
3.3 Numerical results The ability of the schemes in the capture of shock layers for (1) is evaluated when testing their sensitivity in the prediction of the end states with respect to the mesh refinement. The initial data is made of two constant states prescribe by Test A
7 1.4
7r
t*T/V
1.6
0.01
B
1.4
1.6
1
C
1.4
1.6
100
P 1 1.05518 1 1.92678 1 1.01595
u 1 -0.88895 1 -1.25451 1 -0.93415
P 1 0.15031 1 2.74247 1 0.24142
Pr
0.6 0.33869 0.6 2.09561 0.6 0.15528
Problems are directly motivated by the three distinct regimes that underly the flow model under consideration and that are dictated by the amplitude of the viscosity ratio ^T/^- The viscosities uandd /JLT are assumed to be positive constants we referre to [1] for the setting of varying viscosities and non zero heat conductivities, conductivities. In all the benchmarks discussed below, the Reynolds number is set at the constant value Rey = 105.
All the figures assess that the usual numerical strategy (see [7] or [5] concerning the details of this method) grossly fails to restore the correct end states in the three regimes. Turning considering the L2 projection method, the discrete solutions agree with the exact ones only in case A. Such a property no longer holds for problems B and C and consequently large errors occur. These two schemes furthermore suffer from a dramatic sensitivity with respect to mesh refinements for problem C in that discrete solutions do not seem to converge to a given limit function even for the finest proposed grids. By contrast and concerning benchmarks A and B, the discrete solutions stay non sensitive with respect to the mesh refinement but the "limit" function does not coincide with the expected exact solution. Turning considering the nonlinear L2 projection method, it produces approximate solutions that achieve a fairly good agreement with the exact solutions while staying almost non-sensitive with Ax in the three investigated regimes.
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Classical Scheme
L2 Projection Scheme
Figure 1: Problem A : Classical Scheme
Nonlinear Projection Scheme
ur / u < < < 1
L2 Projection Scheme
Figure 2: Problem B
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:: ut / u = 1
Nonlinear Projection Scheme
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Figure 3: Problem C : ur / u> > 1 3.4. Bibliography [1]
[2]
C. BERTHON AND F. COQUEL, Nonlinear projection methods for systems in non conservation form, work in preparation.
G. DAL MASO, P. LEFLOCH AND F. MURAT, Definition and weak stability of a non conservative product, J. Math. Pures Appl, 74, 483-548 1995.
[3]
E. GODLEWSKI AND P.A. RAVIART, Hyperbolic systems of conserva-
tions laws, Applied Mathematical Sciences, Vol 118, Springer 1996. [4]
T. Y. Hou AND P. G. LEFLOCH, Why nonconservative schemes converge to wrong solutions : error analysis, Math, of Comp., Vol 62, No 206, 497-530 1994.
[5]
B. LARROUTUROU AND C. OLIVIER, On the numerical appproximation of the K-eps turbulence model for two dimensional compressible flows, INRIA report, No 1526 1991.
[6]
P.G. LEFLOCH, Entropy weak solutions to nonlinear hyperbolic systems under non conservation form. Comm. Part. Diff. Equa. 13, No 6, 669-727 (1988).
[7]
B. MOHAMMADI AND O. PlRONNEAU, Analysis of the K-Epsilon Turbulence Model, Research in Applied Mathematics, Masson Eds 1994.
[8]
P. A. RAVIART AND L. SAINSAULIEU, A nonconservative hyperbolic system modelling spray dynamics. Part 1. Solution of the Riemann problem, Math. Models Methods in App. Sci., 5, No 3, 297-333 1995.
About a Parallel Multidimensional Upwind Solver for LES
D. Caraeni S. Conway L. Fuchs Division of Fluid Mechanics, Lund Institute of Technology SE-221 00 Lund, Sweden
ABSTRACT A parallel flow solver has been developed at Lund Institute of Technology, for Large Eddy Simulations (LES) of turbulent compressible flows. The numerical algorithm is based on the Residual Distribution scheme approach. The scheme employs an extremely compact stencil, while still having second order accuracy, which makes it well suited for parallelization. In this paper we report about the numerical scheme and about the parallel algorithm implemented in the code. First the performance concerning the parallel scalability is addressed. Finally some results for the LES of the classical channel flow are presented. Key Words: Residual Distribution Scheme, Large Eddy Simulations, Parallel Flow-Solver.
1. Introduction The advances made in computer technology over the last years, have led to a great increase in the engineering problems which can be simulated using CFD. The computation of flows over complex geometries at relatively high Reynolds number is becoming more common using LES, as recent reviews on the subject have shown [PIO 98]. LES for engineering applications are typically extremely expensive, requiring huge resources in terms of processor power and memory, and long computational times. When using large parallel computations, LES becomes accessible for many industrial applications. Accurate numerical algorithms, well suited for parallelization are needed in order to make LES available for production CFD.
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2. LES Equations The expression for the LES equations, can be obtained by "top-hat" Favrefiltering of the time dependent compressible Navier Stokes equations, written in conservative form:
The subgrid terms representing the subgrid stress tensor, rs^s = p[uiUj — UiUj],, the subgrid heat flux, Hj9S = ~p[Euj — EUJ], and the subgrid viscous work, GSj9S = ~f>\TijUii — TijUi], all require modelling. In the present simulations, the r^js was modeled by using a Sub-Grid Scale (SGS) model, as described below. All the other SGS terms presented above, are not explicitly modeled. 3. SGS Models In the Large Eddy simulation technique, the largest scales of the turbulence are resolved and the effect the smallest scales have on the resolved scales, is modeled. This is done by introducing a SGS model. In the present code, three SGS models have been implemented: the Smagorinsky model, Lilly's dynamic model [LIL 91] and the Dynamic Divergence model (DDM)..Thehe DDM model is a novel, anisotropic dynamic SGS model with independently determined coefficients in each coordinate direction. Note that in this case, it is the divergence of the SGS stress tensor, r^,that is modeled [JHF 98] [SCF 98]. For the dynamic models, the model parameter is calculated dynamically at each point from the instantaneous flow field. Negative values of this parameter enable back-scatter, which implies that turbulent energy can be transferred intermittently from the small scales to the larger ones. Local filtering, and artificial bounds, have to be used, in order to avoid large oscillations in the model parameter.
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4. Numerical Algorithm Residual Distribution (RD) or Fluctuation Splitting (FS) schemes date back to the late 80's [ROE 82], when P.L. Roe developed schemes for solving scalar advection-diffusion problems on unstructured meshes. More recently, Van der Weide and H.Deconinck [PDE 97] proposed a matrix generalization of the scalar schemes which can be applied to the solution of non diagonalizable hyperbolic systems. Wood proved [WKK 98] that RD schemes give a better accuracy of the solution, for advection diffusion problems, when compared with the classical finite-volume formulations. Though most applications of RD schemes have been limited so far to steady state computations [DSB 94], we extended their application to accurately simulate time dependent flows [CAO 99] [CAB 99]. Our code uses a second order in time, implicit scheme of Jameson-type [JAM 91]. This is a dual-time step scheme, with sub-iterations to converge the solution at every new real-time step. Multigrid iterations are performed to accelerate the convergence of the solution in these sub-iterations. The code works on an originally developed unstructured mesh, cell-vertex data structure. A hierarchical oct-tree organization of the data was employed with a single tetrahedron division rule. This was shown to allow significant reductions in memory usage, [SCL 97].The discretization of the convective term of the Navier-Stokes equations is based on the matrix Residual Distribution schemes approach [PDE 97]. A finite element, central-Galerkin scheme has been used to discretize the viscous part, the unstationary term and the source terms, i.e. Coriolis forces, centrifugal forces, etc. The update scheme is u™+l
(4)
where U = ( is the conservative variables vector, j = 1..3, AT is the pseudo-time step, n - is the real-time step index, T is one of the cells (tetrahedral) sharing the node i, i — 1..4 is the node index, Vi is the volume of the dual cell i7;, surrounding the i node. Here is the approximation of t/"+1 at the pseudo-time step k, (3f is the modified matrix distribution coefficient for the convective term, iscid advective-residual over the cell T, (5)
where F/ = F?(Un+l>k) is the advective flux vector, and FJ = F?(Un+l>k) is the viscous flux vector, both computed for {Jn+l>k^ nj^ are the cell-face inward normal vectors, ^(^ • id is the contribution of the advective part
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of the Navier-Stokes equations to the nodal residual, ^(FJn^), the contribution of the viscous part of the Navier-Stokes equations to the nodal residual, llli}implicit *s tne volume average of the unstationary term, on the (T n fii). We used an implicit second order discretization with finite differences for [^f]:
A modified second order matrix Residual Distribution scheme has been used for the convective part of the Navier Stokes equations. It employs a dynamic correction of the artificial dissipation of the scheme based on the local flow characteristics. When doing LES, the artificial dissipation of the scheme, i.e. the LW scheme [PDE 97] in our case, is not desired and it can be shut off in regions far from shock waves. Close to shock waves, the artificial dissipation of the original scheme, is reintroduced. If strong shock waves are present, a continuous blend between the second order scheme and a positive first order scheme, i.e. the Narrow scheme, is used in the shock wave region. This enabled us to have monotone shock capturing, while still using very small dissipation in the smooth flow regions. The matrix distribution coefficients, /3/\ are given by:
The matrix distribution coefficients are computed, for the second order scheme, LW:
and for the positive first order, Narrow scheme:
Here vceu is a cell-CFL number. Ki are the Jacobians of the convective dFc flux vector relative to face i, Ki = -Qjj-nj^. The coefficients vcei\, and attend are corrected dynamically. When doing LES, they vanish in regions far from flow field discontinuities. More details about the matrix generalization of the Residual Distribution schemes can be found in [PDE 97]. 5. Parallel Algorithm
Since the Fluctuation Splitting algorithm only needs to access the first order neighbors to update the solution at a vertex, the parallelization process is
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greatly simplified. If it were necessary to access the second-order neighbors, or even higher-order neighbors, then for any vertex that lies along a boundary between processors, a very elaborate and likely expensive method for communicating the appropriate update information across these inter-partition boundaries would need to be devised. Second order classical finite volume codes require access to the second-order neighbors. However, these algorithms get around the complication of having to store the communication information to reach these second order neighbors across processors by using a two-step update procedure. This requires also two communication cycles per time step [VNK 92], whereas the Fluctuation Splitting algorithm only needs one. This makes the fluctuation splitting method very attractive for parallel implementation. The Parallel Virtual Machine (PVM) defacto standard has been used to implement the parallel algorithm in our code [CAD 99]. The code runs as effectively on large distributed memory machines (Cray T3E, IBM SP2...), on the SGI Origin 2000 servers or on a heterogenous collection of platforms. The code takes advantage of its unstructured grid for realizing an almost perfect load balancing by assigning a number of volumes to compute to each processor, proportional with the processors speed. The parallelization of the code has been done using the SPMD paradigm, i.e. the same code is executed in all processors. The global data structure, i.e. grid, field variables, etc. is decomposed into a number of subdomains, equal with the number of running tasks. While any algorithm for domain decomposition can be employed, so far only one dimensional domain decomposition has been used. This is done in a separate preprocessing step and does not affect the solvers flexibility. Each task runs with its own data structure, which is a disjunct part of the global data structure. The tasks are implicitly synchronized by using blocked data exchange. The algorithm implies at each real time step sub-iterations for converging the solution. In each sub-iteration, the solver: - computes the residual, i.e. the integral of the convective, diffusive, source term and unstationary part of the Navier-Stokes equation, over one tetrahedral cell, looping through all cells. Splits the residual according with the distribution scheme in fluctuations, and sends fluctuations to the nodes, where these are collected in some special fields. - communicates using PVM, with neighboring parallel processes, the fluctuations accumulated in nodes and some other useful information, for nodes laying on the domains interface. - imposes boundary conditions, - loops through all nodes, to update the nodal values using the fluctuations stored in those special fields. As it can be seen, the algorithm needs no communication while computing and distributing the fluctuations. This step in the solver algorithm requires the longest computational time. It requires only one back and forward communication for each sub-iteration before applying the boundary conditions and updating the nodal values. The volume size of this communication is reduced, since only nodes on the domains interface exchange information.
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6. Numerical Results LES simulations of complex turbulent flows in engineering applications, using the present code, have been presented in [CAO 99] [CAB 99]. Here we only address the parallel scalability of code and the accuracy of our LES computations.
6.1. Parallel scalability The code has been run on an SGI Origin 2000 platform, on up to 8 processors, and also on a cluster of 2 WinNT PC's, using an Ethernet connection. The results, addressing the parallel speed-up are presented graphically in figure 1. It can be seen that the performance, when using the two WinNT PC cluster, has been close to 75%, due to poor communication while using the Ethernet connection. The code proved instead to have excellent scalability on the Origin platform (98%) [CAD 99].
6.2. LES of channel flow Results for the classical turbulent channel flow are presented. The Reynolds number based on the bulk velocity and the channel height, is 5800. No-slip isothermal boundary conditions have been set at the walls, while periodicity boundary conditions have been used both in the streamwise (x) and the spanwise (z) directions. Computations have been performed using the dynamic DDM model. The grid size is (2-7r<5) (6) (2?r<5/3) where 8 is the channel height. The grid has (85)(51)(51) nodes and aprox. 1,240,000 tetrahedral cells. The grid is stretched in the y—direction (yl = <5(1 — cos(/?;))/2, for pi = 7r(j — l)/(N — 1), j = 1, 2,...., N. Here N is the number of grid points in the y—direction).A uniform grid has been used in the streamwise and the spanwise directions. The flow has been simulated for enough flow-through times (the domain length in the streamwise direction divided by the bulk velocity) to obtain a statistically stationary turbulent channel flow. Data for statistics are accumulated over the last 15 flow-through times. Planar averages are calculated by averaging for all points on planes parallel to the walls and in time, and the results are presented as a function of wall-distance (y) only, and are compared with experiments (Kreplin, 1979) and with DNS data (Kim, 1987). The averaged non-dimensional friction velocity UT = 0.06156 (i.e. uT/Ub, where Ub is the bulk velocity) compares well with the DNS and the experimental (i.e. UT = 0.0643) result. Figures 2 to 5 show planar averaged time averages of the mean of the axial velocity and Root-Mean-Square of velocity fluctuations (RMS of Favre U",V",W"). The results are normalized using the friction velocity, UT. The results obtained using the DDM model compare well with the DNS and the experimental results. Figure 6 shows an instantaneous dis-
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tribution of the skin friction coefficient, C/.
7. Future Work Future work will include extensive testing of different dynamic SGS models. The goal is to perform parallel LES for complex engineering applications.
[PIO 98]
PlOMELLl, U., Large Eddy Simulation: present state and future directions
[ROE 82]
ROE, P.L., Fluctuations and signals. A framework for numerical evolution problems, K.W. Morton and M.J. Baines, editors, Numerical Methods for Fluid Dynamics, Academic Press,
1982. [PDE 97]
PAILLERE, H. et al., Upwind residual distribution methods for compressible flows : An alternative for finite volume and finite element methods, VKI 28th CFD Lecture Series, March 1997.
[DSB 94]
DECONINCK, H. et al., High resolution shock capturing cell vertex advection schemes on unstructured grids, VKI Lecture Series, March 21-25 , 1994.
[WKK 98]
WOOD, WILLIAM A. et al., Diffusion characteristics of upwind schemes on unstructured Triangulations, AIAA 98-2443, Albuquerque, NM 1998.
[JAM 91]
JAMESON, A., Time dependent computations using multigrid, with applications to unsteady flows past airfoils and wings, AIAA paper, 91-1596, 1991.
[CAB 99]
CARAENI, D. et al., LES of spray in compressible flows on Unstructured Grids, AIAA 99-3762, Norfolk, 1999.
[CAO 99]
CARAENI, D. et al., Large Eddy Simulation of the Flow in a Bladed Diffuser,r, TSFP, Santa Barbara, 1999
[CAD 99]
CARAENI, D. et al., Parallel NAS3D: An efficient algorithm for engineering spray simulations using LES, pres. Cl-C, International Parallel CFD'99, 1999, WiUiamsburg.
[SCL 97]
MlTRAN, S. et al., Large Eddy Simulation of rotor stator interaction in centrifugal impeller, JPC, Seattle, July 1997.
[JHF 98]
HELD, J. et al., Large Eddy Simulation of separated transonic flows around a wing section, AIAA 98-0405, Reno 1998.
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[SCF 98]
CONWAY, S. et al., Investigation of the flow across a swirl generator using LES, AIAA 98-0921, Reno 1998.
[LIL 91]
LILLY, D.K., A proposed modification of the Germano subgridscale closure method, Phys. Fluids A, 4:633-635, 1991.
[VNK 92]
VENKATAKRISHNAN V. et al., A MIMD implementation of a parallel Euler solver for unstructured grids, The Journal of Supercomputing, Vol. 6, 1992.
Fig. 1 Parallel SpeedUp
Fig 2. Planar Time Averages of U
Fig 3. Planar averages of U"rms
Fig 4. Planar averages of V'rms
Fig 5. Planar averages of W'rms
Fig. 6 Skin friction distribution
A higher-order-accurate upwind method for 2D compressible flows on cell-vertex unstructured grids
L. A. Catalano Istituto di Macchine ed Energetica Politecnico di Bari Via Re David 200, 70125 Bari, ITALY E-mail: [email protected]
ABSTRACT A finite-volume method for the solution of two-dimensional inviscid compressible flows on cell-vertex unstructured grids is presented. The method is based on a novel bi-linear reconstruction of the unknowns and on a standard flux-difference-splitting scheme. Moreover, a new approach is proposed to achieve the same higher-order accuracy also near solid walls. The method is validated by computing the inviscid flow in a two-dimensional cascade in subsonic and transonic conditions. Key Words: higher-order reconstruction, upwind, unstructured.
1. Introduction In the last decade, a great effort has been devoted by many CFD researchers to the development of higher-order-accurate upwind solvers on unstructured grids. Different approaches have been proposed, including both finite-element and finite-volume discretizations [BAR 91, DEC 92, CAT 97, HAL 97, SEL 96], the major difficulty being the discretization of the inviscid terms in the conservation equations. Concerning the finite volume discretization, most of the upwind schemes proposed to date are based on a gradient-based reconstruction of the flow variables onto the two sides of the surface which defines the finite volume built around each node. Then, an approximate Riemann solver is applied at each interface to select the upwind contributions.
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How to perform a higher-order reconstruction on an unstructured triangular grid is not trivial: if the unknowns are located in the center of each triangle, the finite volume is the cell itself, and the higher-order reconstruction onto its sides becomes a very cumbersome task, because of the difficulty of defining and computing the flow gradients. For such a reason, many researchers have redirect their efforts to cell-vertex discretizations: a dual mesh is first constructed by tracing the medians of each cell, so as to build a finite volume around each node [BAR 93, SEL 96, HAL 97]. Clearly, the interfaces are now located at the midpoint of each side. To knowledge of the author, all of these methods base the reconstruction on a nodal definition of the flow variable gradient. Different approaches have been proposed to compute its value from the gradients in the surrounding cells, but all of them appear complex and time-consuming, see e.g. [BAR 93, HAL 97]. In this paper, an alternative, much simpler approach is proposed, which also includes a new higher-order near-wall discretization. 2. Numerical method 2.1. Governing equations The governing equations for two-dimensional, compressible, inviscid flows, are written in integral form as: (1)
In eq. 1, n is the inward normal of the contour of 5, dS, U = (p, pu, pv, pe°)T is the vector of the conservative variables and F-n = [(pvn), (puvn+pnx), (pvvn + pny), (ph°Vn)]T is the flux entering through the unit length of dS. As usual, p is the density, p is the pressure, e° is the total internal energy and h° is the total enthalpy. Moreover, v will denote the velocity vector, with normal and tangential components vn — v • n and vs = v • s, respectively, and with Cartesian components u and v. The system of governing equations is closed by assuming perfect gas.
2.2. Space discretization The domain is discretized by means of an unstructured mesh composed of triangles with unknows located at each cell-vertex. In particular, the primitive variable vector Q — (p,u,v,p)T is chosen as unknown. A finite volume is constructed around each internal node by connecting the barycenters of two neighbouring triangles, see the node i in fig. 1. An upwind discretization of the RHS of eq. 1 is then obtained as follows: a left state and a right state are reconstructed at the interface (ij] defined on each side connecting the node i
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interface (ij)
auxiliary cells Figure 1: Construction of finite volumes for internal and boundary nodes. Determination of the cell Cji. Construction of the auxiliary cells. with each sorrounding node j. Then, a standard Riemann solver is used to select the upwind contributions to the flux. The left and the right states can be accurately reconstructed by employing a bi-linear variation of Q: (2)
In eq. 2, Iji arid are the two opposite vectors pointing from the two nodes to the intersection of the interface with the side, see fig. 1. Concerning the definition of the gradients, first consider the interface (i + 1/2) of a uniform one-dimensional grid; in such a case, Qf+1/2 is linearly reconstructed as: (3)
It must be remarked that the reconstruction is based on the gradient of Q in the left-neighbouring cell, rather than on the gradient defined in the node i. Similarly, in two dimensions, the gradient (^Q)ji (similar arguments hold for (VQ)jj) must be defined in one of the cells sharing the node j, rather than in the node itself, as it suggested in [BAR 93, HAL 97]. The extension of the previous arguments to two dimensions suggests to define (VQ)ji as the gradient in the cell Cji which contains the prolongation of the side (ji), plotted as a dot-dashed line in fig. 1. Clearly, the choice of a cell-vertex triangular grid allows to define a bi-linear variation of Q in each cell, thus defining the cell gradient (VQ)ji = (VQ}Cji uniquely: (4)
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In eq. 4, Qf. is the primitive variable vector in the node k. Moreover, the opposite side k has length s^ and inward normal n/t. Standard one-dimensional limiters can also be applied to the gradient, but have not been introduced yet. The flux-difference-splitting of Roe [ROE 86] is then used to solve the Riemann problem defined at each interface. The flux is computed as (5)
In eq. 5, 0:^, k = 1, ...,4, are the intensities of the entropy, of the shear and of the two acoustic waves, and A&, k = I,..., 4, are the corresponding propagation velocities:
(6)
Finally, Cfc, k — 1,..., 4 are the eigenvectors which project each wave contribution onto the conservative variable vector:
(7)
In eqq. 6 and 7, <5( ) = ()# — ( )L, and ~ denotes the Roe averages:
(8)
2.2. Boundary conditions When a side lies on a boundary, the construction of the finite volumes is completed by tracing one third of the corresponding median and by adding half of the side to the finite-volume contours of its two nodes: as an example, fig. 1 shows the resulting finite volume associated to the boundary node b. Clearly,
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the flux through each boundary side is computed directly by means of the nodal values; moreover, solid boundaries need only to add the corresponding pressure forces to the momentum equation. Characteristic boundary conditions are then applied to the nodal residuals to ensure that physical boundary conditions are satisfied. The lack of further cells beyond the boundaries make impossible to perform a higher-order reconstruction onto the near-boundary interfaces. The reduced
Figure 2: Computational mesh. accuracy is negligible for far-field boundaries, but a significant amount of numerical entropy would be generated near solid walls. A simple procedure to overcome this problem is here proposed: a row of auxiliary cells is created by adding an isosceles triangle beyond each solid face and by connecting the corresponding auxiliary nodes, see the dashed cells in fig. 1. The states in the auxiliary nodes are updated by imposing the following conditions of Isentropic Simple Radial Equilibrium at the mid-point of the solid face:
(9)
Rc being the radius of curvature at wall. 2.3. Time integration The state in each node is updated by means of a two-stage Runge-Kutta explicit scheme with non-optimal coefficients 0.4 and 1 and CFL number 0.35.
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3. Results The method previously described has been applied to the computation of the inviscid flow in a cascade of VKI LS-59 gas turbine rotor blades, for two different values of the outlet pressure. The blade profile has been modified at the trailing edge by adding an artificial wedge (having no load) in order to simulate the presence of the recirculation zone, which cannot be captured by the present inviscid formulation. Fig. 2 shows the grid used in both the cases analyzed, obtained by slightly modifying a structured grid with 129 x 17 nodes (97 nodes are located on the blade surface). Clearly, the resulting quality of the grid is rather poor. The Mach number contours obtained for outlet Mis
Figure 3: Mach number contours for outlet Mis = 0.81 (AM = 0.05,). (isentropic Mach number) equal to 0.81 are shown in fig. 3. A good agreement
Figure 4: Experimental and computed distributions of Mis on the blade for outlet Mis =0.81. between the computed distribution of Mis on the blade and the experimental one [KIO 86] is visible in fig. 4.
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Figure 5: Mach number contours for outlet M{S — 1 ("AM = 0.05J. The transonic flow with outlet MJS equal to 1 has been then considered: the Mach number contours in fig. 5 show the formation of two shocks on the suction side of the blade, which are both sharply captured. Small oscillations are present and can be eliminated by applying a standard limiter. The surface Mach number distribution shown in fig. 6 is again in good agreement with the experimental data of [KIO 86], except for the discrepancy on the suction side of the blade, due to the presence of a separation bubble caused by the shockboundary layer interaction. Such a phenomenon is obviously missed by the present inviscid analysis.
Figure 6: Experimental and computed distributions of MIS on the blade for outlet Mis = 1-
3. Conclusions A finite-volume method for the solution of two-dimensional inviscid com-
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pressible flows on cell-vertex unstructured grids has been presented. In particular, a novel approach to the higher-order reconstruction of the unknowns has been proposed, and extended to the near-wall regions by means of a row of auxiliary cells. The method has been validated by computing the inviscid flow in a two-dimensional turbine cascade in subsonic and transonic conditions. The extension to the discretization of the viscous-flow conservation equations can be based on state-of-the-art methodologies. 4. Bibliography
[BAR 91]
BARTH T. J., Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations, Lecture Series 1991-06, Von Karman Institute, 1990.
[BAR 93]
BARTH T. J., A 3-D least-squares upwind Euler solver for unstructured meshes, Lecture Notes in Physics, 414, Springer Verlag, pp. 90-94, 1993.
[CAT 97]
CATALANO L. A. et a/., Genuinely multidimensional upwind methods for accurate and efficient solutions of compressible flows, Euler and Navier-Stokes solvers using multidimensional upwind schemes and multigrid acceleration, Notes on Numerical Fluid Mechanics, 57 Vieweg, Braunschweig, Germany,, 1997.
[DEC 92]
DECONINCK H. et a/., Multidimensional upwind methods for unstructured grids, Agard R-787, May, 1992.
[HAL 97]
HALLO L. et a/., An implicit mixed finite-volume-finiteelement method for solving 3D turbulent compressible flows, International Journal for Numerical Methods in Fluids, 25, pp. 1241-1261, 1997.
[KIO 86]
KlOCK R. et a/., The transonic flow through a plane turbine cascade as measured in four European wind tunnels, Transactions of the ASME, Journal of Engineering for gas turbines and power, 108, No. 2, pp. 277-284, 1986.
[ROE 86]
ROE P. L., Characteristic based schemes for the Euler equations, Ann. Rev. Fluid Mech., 18, pp. 337-365, 1986.
[SEL 96]
SELMIN V., FORMAGGIA L., Unified construction of finite element and finite volume discretizations for compressible flows, International Journal for Numerical Methods in Engineering, 39, pp. 1-32, 1996.
A New Upwind Least Squares Finite Difference Scheme (LSFD-U) for Euler Equations of Gas Dynamics N. Balakrishnan
Praveen. C
Assistant Professor Graduate Student CFD Centre, Department of Aerospace Engineering Indian Institute of Science, Bangalore - 560 012 ABSTRACT A new upwind Least Squares Finite Difference Scheme(LSFD-\J) has been developed. The fundamental principle underlying this method is the representation of the derivatives of the fluxes appearing in the conservation laws, using a Generalised Finite Difference strategy based on the method of least squares. This method can operate upon any kind of grid and requires only local connectivity information pertaining to a cloud of grid points around any given node. The results obtained are very encouraging and the use of LSFD-U in the computation of flows past complex configurations is extremely promising. Key Words: Finite difference, Finite volume, Upwind schemes, Hyperbolic equations, Method of Least Squares. 1. Introduction One of the remarkable progresses made in the area of CFD, in recent years is the development of Grid Free Method [1-4] for numerically solving the conservation laws encountered in fluid dynamics. The fundamental principle underlying this method is the representation of the derivatives of the fluxes appearing in the conservation laws using a generalised finite difference strategy based on the method of least squares. This method which can operate upon any kind of grid (structured, unstructured or cartesian) requires only local connectivity information pertaining to a cloud of grid points around any given node. The utility of this method in computing flow past complex configurations is extremely promising. The present work draws its inspiration from the fact that this method has been applied only in the framework of flux vector splitting schemes and not in the framework of flux difference splitting schemes. Here we have attempted to extend the applicability of this Grid Free Method to the framework of flux difference splitting schemes and in the process arrived at an entirely new methodology equally applicable to flux vector splitting schemes. In section 2, we present briefly the details regarding the Least Squares Kinetic Upwind Method (LSKUM) [1-4]. In section 3, the new least squares scheme is presented with a brief review of the flux difference splitting schemes. In section 4, we present the difficulties in extending the methodology to 2D and 3D flows, and also present two variations of the scheme, which would circumvent such problems. In section 5, we present the results and discussions. Concluding remarks are made in section 6. 2. Least Squares Kinetic Upwind Method (LSKUM) The kinetic schemes for solving Euler equations of gas dynamics are obtained by exploiting the fact that these equations are moments of the Boltzmann equation of kinetic theory for gases. Consider the ID Boltzmann equation : f, + vf x = 0 (1) where f is the velocity distribution function which is a Maxwellian and v is the molecular velocity. The fundamental principle underlying LSKUM is that the discrete approximation to f x appearing in the Boltzmann equation is obtained using a least squares approximation given
by,
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(2)
where Afj=fj-f 0 ; AXj=Xj-x0, subscript V denotes the node under consideration and 'j' its neighbour. The moment of the discretized Boltzmann equation will lead to an upwind scheme for the Euler equation, if the stencil of the grid points to be used in equation (2) is chosen, taking into account the direction of signal propagation. In other words, discrete approximation to fx at any given point is obtained by using the grid points on its left if v>0 and vice versa. An interested reader is referred to the papers cited above for a number of interesting developments in LSKUM, such as the use of weighted least squares in equation (2). This idea when extended to the flux vector split framework of Euler equations, given by, Ut+Fx++Fx- = 0 (3) where U is the vector of conserved variables and F is the flux vector, the discrete approximation to F x *at any given point will involve grid points to its left, and Fx~ will involve grid points to its right. 3. Upwind Least Squares Finite Difference Method (LSFD-U) Inspired by the fact that the discrete least squares approximation to the derivative Fx involves the flux difference term AF, it was thought that it would be appropriate to make use of flux difference splitting in the least squares framework. Before we discuss the details of the present least squares algorithm, we briefly discuss the flux difference splitting scheme as applied to finite volume framework.
Fig 1.
Typical ID Finite Volume Computational Domain
Fig 1.depicts a typical ID finite volume computational domain. In the flux difference splitting scheme[5], the total flux difference AF = FR - FL is split into a positive part (AF)+ corresponding to the right running waves and a negative part (AF)" corresponding to the left running waves, based on a suitable linearization procedure, in such a way that the interfacial flux F( is given by (4)
In a finite volume framework, an interfacial flux obtained as an average of the above two expressions is made use of in the state update formula. Now we describe the present methodology. Equation 4 given above clearly suggests that the flux difference between any fictitious interface perpendicularly intersecting the line connecting the two grid points and the points themselves, can be obtained using a suitable linearization procedure. At the heart of the present methodology is the use of such flux differences based on upwinding principle in the discrete least squares approximation to Fx appearing in the Euler equations. This leads to an upwind scheme based on the least squares principle. Also, it is not necessary that the present methodology should be used only in conjunction with Flux Difference Splitting Schemes. The very fact that an upwind estimate of
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the flux at the fictitious interface is all that is required for the determination of flux differences involved in the present method, suggests the use of this method in conjunction with all upwind flux formula. As can be easily seen, the present methodology makes use of a global stencil of grid points in contrast to the methodology described in section 2, which requires an upwind stencil of grid points. Also, we make use of an upwind estimate of the interfacial flux for determining AF's appearing in the least squares formula, in contrast to the explicit use of the nodal values of the fluxes in the earlier framework. 4. LSFD-U in 2D Here we present some of the interesting problems we face when extending the algorithm just described for solving 2D flows. Fig.2 gives a typical 2D stencil of grid points. Let Ij represent the fictitious interface drawn across the line connecting the point under consideration 'o', and its jth neighbour, and n : represent the unit vector along
'oj'.
Representing the flux along oj by
we have (5) (6)
where F and G are fluxes in x and y directions respectively. Now our job reduces to recovering the information regarding the gradients of the 2D fluxes namely VF0 and VG fr°m tne many ID flux difference terms given in equation 6. The derivatives Fx and Gy thus recovered would eventually be used in the state update formula. Equation 6 represents an over determined system of equations and it appears that the straight forward way to obtain the gradients of the fluxes is to minimise X E • w^h respect to the derivatives of the fluxes, where, (7)
Unfortunately, simple algebra would demonstrate that such a procedure leads to a singular system. To circumvent this problem we suggest the following two methodologies. 4.1 Method 1 Method 1 draws its inspiration from the work of Ghosh and Deshpande [2]. Here we effect locally a co-ordinate transformation (x,y)->(^,T|), in such a way that one of the axes (Q coincides with the streamwise direction. It is a well known fact that the fluxes normal to the streamwise co-ordinate direction involve only pressure terms and a global stencil of grid points can be used for approximating the derivatives of such fluxes without loss in stability. If F and G represent the fluxes along the new co-ordinate directions E, and r\ respectively, the derivatives Gc and G can be obtained using the least squares procedure described in [3], S T\ making use of a global stencil of grid points. Thus the streamwise rotation of the co-ordinates leaves us with a non singular system involving Ft and Fn • A simple least squares procedure described in the previous section can be adopted for the estimation of R and R,. The derivatives R and G thus determined are substituted in the state update formula. It should
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be remarked that in this method, the second and third components of the fluxes F and G represent ^ and T] momentum conservation. 4.2 Method 2 In this method we locally rotate the co-ordinate system from (x,y) —^ (^,r|),
m
such a way that p + G& = 0- where F and G represent the fluxes along the new coordinate directions £, and r\ respectively. Note that the second and third components of the flux vectors Fand G still represent the x and y momentum conservation. It can easily be demonstrated that a co-ordinate system rotated at an angle p\ given by, (8)
would satisfy the condition that p + Gt = 0 • Th£ gradients of the fluxes used in the estimation of (5 are obtained using the least squares procedure [3] making use of a global stencil of grid points. This results in an overdetermined system of equations with two unknowns Ft and G~ , which can be solved using least squares procedure. The derivatives thus determined are used in the state update formula. 5. Results and Discussions The new least squares upwind finite difference method (LSFD-U) is validated using standard ID and 2D test problems. In the computations high resolution is obtained based on a linear reconstruction procedure [6]. Non physical oscillations in the solution are suppressed using Yenkatakrishnan limiter [7]. In all the computations presented in this work the fictitious interface is always placed at the mid point of the line segment under consideration. Figure 3 gives the results obtained for the ID shock tube problem of Sod [8]. The results are obtained on a non-uniform grid generated using cosine spacing for grid points. One hundred grid points have been used in the computation. Roe [5] flux has been used in these calculations. The grids made use of in the 2D computations are presented in Figure 4. The grid details are given in Table 1. All the 2D computations have been made with KFVS [9] flux formula.
Configuration Cylinder NACA0012
Table 1 No. of nodes 4317 4733
No. of nodes on the wall 160 160
The pressure contours obtained for a low subsonic flow past a cylinder(in the incompressible limit; M«=0.1) are presented in Figures 5 and 6. The wall pressure data are compared with the exact potential flow solution in Figure 7. The Mach contours obtained for subsonic(Mro=0.63, angle of attack=2°) and transonic (MM=0.85, angle of attack=l°) flows past NACA0012 airfoil are presented in Figures 8,9,11 and 12. The CP values are compared with those obtained using the cell vertex finite volume scheme in Figures 10 and 13. The first order results obtained using Method 2 are identical to those obtained using Method 1, and therefore are not presented here. The CL and CD values obtained using LSFD-U are compared with GAMM[10] results in Table 2.
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Table 2 M.=0.63, AOA=2°
Method
I II GAMM
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M«=0.85, AOA=1°
CL
CD
CL
CD
0.3387 0.3343 0.3335
0.0014 0.0006 0.0000
0.3772 0.3806 0.3790
0.0550 0.0545 0.0576
From the results it is evident that the new LSFD-U framework is capable of capturing all features expected out of inviscid compressible flows. Also, it should be remarked that the linear reconstruction procedure developed for the finite volume schemes, when used in conjunction with the present LSFD-U framework, can produce excellent improvement in solution resolution. 6. Conclusions A new upwind least squares finite difference method has been developed. The new scheme by the virtue of using a least squares framework can be considered as a Grid Free Method. It has an added advantage of making use of a global stencil. The way the interfacial fluxes are calculated in the new scheme resembles that of finite volume method and therefore all the developments that have taken place in finite volume method, like the method of reconstruction [6] can be directly adopted. The use of LSFD-U in the computation of flows past complex configurations is extremely promising.
Acknowledgements The authors like to thank Mr J. C. Mathur. NAL, Bangalore, for providing the unstructured mesh generator. The authors like to thank Mr. Krishnakumar, under graduate student, 1IT, Kharagpur, for his involvement in the initial phases of development of the code used in the computations, during his stay at IISc, Bangalore, as a Summer trainee. The authors also like to express their gratitude to Mr. Harish. R, project assistant, CFD Centre, IISc, for his invaluable help in preparing the manuscript. References S.M. Deshpande, A.K. Ghosh and J.C Mandal, "Least Squares Weak Upwind Method for Euler Equations", 89 FM 4, Fluid Mechanics Report, Department of Aerospace Engineering, Indian Institute of Science, Bangalore. 2. A.K. Ghosh, "Robust Least Squares Kinetic Upwind Method for Inviscid Compressible Flows", June 1996, PhD thesis. 3. S.M. Deshpande, P.S. Kulkarni, and A.K. Ghosh, "New Developments in Kinetic Schemes", Computers Math. Applic., Vol 35, No 1/2, pp. 75-93, 1998 4. K. Anandhanarayanan, D.B. Dhokrikar, V. Ramesh and S.M. Deshpande, "A Grid Free Method for 2D Euler Computations using Least Squares Kinetic Upwind Method", Proceedings of the third Asian Computational Fluid Dynamics Conference, pp. 390, vol 2, December?"1 - 1 Ilh 1998, Bangalore, India. 5. Roe, Philip. L. "Approximate Reimann Solvers Parameter Vectors and Difference Schemes", Journal of Computational Physics, Vol. 43, pp. 357-372, 1981 6. T.J. Barth " Higher Order Solution of the Euler Equations on Unstructured Grids using Quadratic Reconstruction", AIAA-90-0013, 1990 7. V. Venkatakrishnan, "Convergence of Steady State Solutions of Euler Equations on Unstructured Grids with Limiters", JCP, vol 118, pp. 120, 1995. 8. G.A. Sod, "A Survey of Several Finite Difference Methods for system of Nonlinear Hyperbolic Conservation Laws", JCP, vol 27, pp. 1, 1978. 9. J.C. Mandal and S.M. Deshpande, " Kinetic Flux Vector Splitting for Euler Equations", Computers and Fluids, vol 23, No 2, pp 447-478. 10. GAMM Workshop on Numerical solutions of Compressible Euler Flows, June 1986. 1.
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Fig 2. Typical 2D grid distribution for LSFD-U
Fig 3. LSFD-U applied to shock tube problem
Fig 4. Unstructured grid used for 2D computations Linear Reconstruction
Fig 5 Pressure contours obtained using Method 1 for M»=0.1
Linear Reconstruction
Fig 6 Pressure contours obtained using Method 2 for M>=0.1
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Fig 7 Pressure distribution over a cylinder at M«,=0.1
Fig 8 Mach contours using Method 1; M«=0.63 and AOA=2°
Fig 9 Mach contours using Method 2 for M^O.63 and AOA=2°
Fig 10 Pressure distribution for M«=0.63 and AOA=2°
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Fig 11 Mach contours using Method I; MM=0.85 and AOA=1°
Fig 12 Mach contours using Method 2 for M^O.85 and AOA=1°
Fig 13 Pressure distribution for M»=0.85 and AOA=1°
A finite-volume algorithm for all speed flows F. Moukalled and M. Darwish American University of Beirut, Faculty of Engineering & Architecture, Mechanical Engineering Department, P.O.Box 11-0236, Beirut, Lebanon. ABSTRACT. A new collocated finite volume-based solution procedure for predicting viscous compressible and incompressible flows is presented. The technique is equally applicable in the subsonic, transonic, and supersonic regimes. Pressure is selected as a dependent variable in preference to density because changes in pressure are significant at all speeds as opposed to variations in density which become very small at low Mach numbers. The newly developed algorithm has two new features; (i) the use of the Normalized Variable and Space Formulation methodology to bound the convective fluxes; and (ii) the use of a HighResolution scheme in calculating interface density values to enhance the shock capturing property of the algorithm. Keywords: Pressure-based method, All speed flows, High-Resolution algorithm.
1. Introduction In Computational Fluid Dynamics (CFD) a great research effort has been devoted to the development of accurate and efficient numerical algorithms suitable for solving flows in the various Reynolds and Mach number regimes. The type of convection scheme to be used in a given application depends on the value of Reynolds number. On the other hand, the Mach number value dictates the type of algorithm to be utilized in the solution procedure. These algorithms can be classified into two groups: density-based methods and pressure-based methods, with the former used for high Mach number flows, and the latter for low Mach number flows. The ultimate goal however, is to develop a unified algorithm capable of solving flow problems in the various Reynolds and Mach number regimes. To understand the difficulty associated with the design of such an algorithm, it is important to understand the role of pressure in compressible flow [KAR 86]. In the low Mach number limit where density becomes constant, the role of pressure is to act on velocity through continuity so that conservation of mass is satisfied. Obviously, for low speed flows, the pressure gradient needed to drive the velocities through momentum conservation is of such magnitude that the density is not affected significantly and the flow can be considered nearly incompressible. Hence, density and pressure are very weakly related. As a result, the continuity equation is decoupled from the momentum equations and can no longer be considered as the equation for density. Rather, it acts as a constraint on the velocity field. Thus, for a sequential solution of the equations, it is necessary to devise a mechanism to couple the continuity and momentum equations through the pressure field. In the
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hypersonic limit where variations in velocity become relatively small as compared to the velocity itself, the changes in pressure do significantly affect density. In this limit, the pressure can be viewed to act on density alone through the equation of state so that mass conservation is satisfied [K.AR 86] and the continuity equation can be viewed as the equation for density. The above discussion reveals that for any numerical method to be capable of predicting both incompressible and compressible fluid flows the pressure should always be allowed to play its dual role and to act on both velocity and density to satisfy continuity. Several researchers [KAR 86, RHI 86, MAR 94, DEM 93, LIE 93] have worked on extending the range of pressure-based methods to high Mach numbers. In most of the published work the first order upwind scheme is used to interpolate for density, exception being in the work presented in [DEM 93] where a central difference scheme blended with the upwind scheme is used. The bleeding relies on a factor varying between 0 and 1, which is problem dependent and has to be adjusted to eliminate oscillation or to promote convergence. In the work presented in [LIE 93] the retarded density concept is utilized in calculating the density at the control volume faces. This concept is based on factors that are also problem dependent and requires the addition of some artificial dissipation to stabilize the algorithm (second-order terms were introduced), which complicate its use. To this end, the objective of this paper is to present a newly developed collocated pressure-based solution procedure that is equally valid at all Reynolds and Mach number values. The algorithm will have two new features. The first one is the use of the Normalized Variable Formulation (NVF) [LEO 87] and/or the Normalized Variable and Space Formulation (NVSF) [DAR 94] methodology in the discretization of the convective terms. The second one, is the use of HighResolution (HR) schemes in the interpolation of density in the source of the pressure correction equation and the convective fluxes in order to enhance the shock capturing capability of the method. The increase in accuracy with the use of HR schemes for density is demonstrated by comparing predictions, for the flow over a bump, obtained using the third-order SMART scheme for all variables except density (for which the Upwind scheme is used) against another set of results obtained using the SMART scheme for all variables including density. 2. Finite volume discretization of the transport equations The conservation equations governing two-dimensional compressible flow problems may be expressed in the following general form:
where (j) is any dependent variable, v is the velocity vector, and p, F*, and Q* are the density (=P/RT), diffusivity, and source terms, respectively. Integrating the above equation over a control volume (Fig. 1) and applying the divergence theorem, the following discretized equation is obtained:
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where J f represents the total flux of <j> across face T and is given by Each of the surface fluxes Jf contains a convective contribution, JJ? , and a diffusive contribution, jj?, hence: where The diffusive flux at the control volume face 'f is discretized using a linear symmetric interpolation profile so as to write the gradient as a function of the neighboring grid points. The convective flux across face f can be written as: where Cf is the convective flux coefficient at cell face T. As can be seen from [6] the accuracy of the control volume solution for the convective scalar flux depends on the proper estimation of the face value <j)f as a function of the neighboring § nodes values. Using some assumed interpolation profile, (j>f can be explicitly formulated in terms of its node values by a functional relationship of the form: where <|)nb denotes the neighboring node (j> values. After substituting [7] into [6] for each cell face and using the resulting equation along with the discretized form of the diffusive flux, [2] is transformed after some algebraic manipulations into the following discretized equation:
where the coefficients ap and a NB depend on the selected scheme and bp is the source term of the discretized equation. \
V
I
Figure 1. Control volume. 3. The NVSF methodology for constructing HR schemes As mentioned earlier, the discretization of the convection flux is not straightforward and requires additional attention. Since the intention is to develop a high-resolution algorithm, the highly diffusive first order UPWIND scheme [PAT 81] is excluded. As such, a high order interpolation profile is sought. The difficulties associated with the use of such profiles stem from the conflicting requirements of accuracy, stability, and boundedness. Solutions predicted with high order profiles tend to provoke oscillations in the solution. To suppress these
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oscillations, the composite flux limiter method [LEO 88] is adopted here. The formulation of high-resolution flux limiter schemes on uniform grid has recently been generalized in [LEO 88] through the Normalized Variable Formulation (NVF) methodology and on non-uniform grid in [DAR 94] through the Normalized Variable and Space Formulation (NVSF) methodology. To the authors' knowledge, the NVSF formulation has never been used to bound the convection flux in compressible flows. It is an objective of this work to extend the applicability of this technique to compressible flows. For more details the reader is referred to [DAR 94]. 4. High resolution algorithm The need for a solution algorithm arises in the simulation of flow problems because a scalar equation does not exist for pressure. Hence, if a segregated approach is to be adopted, coupling between the u, v, p, and P primitive variables in the continuity and momentum equations will be required. The segregated algorithm adopted in this work is the SIMPLE algorithm [PAT 81], which involves a predictor and a corrector step. In the predictor step, the velocity field is calculated based on a guessed or estimated pressure field. In the corrector step, a pressure (or a pressure-correction) equation is derived and solved. Then, the variation in the pressure field is accounted for within the momentum equations by corrections to the velocity and density fields. Thus, the velocity, density, and pressure fields are driven, iteratively, to better satisfying the momentum and continuity equations simultaneously and convergence is achieved by repeatedly applying the above-described procedure. The key step in deriving the pressure-correction equation is to notice that in the predictor stage a guessed or estimated pressure field from the previous iteration, denoted by P , is substituted into the momentum equations, the resulting velocity field, denoted by v , which now satisfies the momentum equations, will, in general, not satisfy the continuity equation. Thus, a correction is needed in order to obtain a velocity and pressure fields that satisfy both equations. Denoting the pressure, velocity, and density corrections by P', v'(u', v'), and p', respectively, the corrected fields are given by: Combining momentum and continuity and substituting P, v, and p using [9], the final form of the pressure-correction equation is: where
From [11] it is clear that the starred continuity equation appears as a source term in the pressure correction equation. Moreover, in a pressure-based algorithm, the pressure-correction equation is the most important equation that gives the pressure, upon which all other variables are dependent. Therefore, the solution accuracy depends on the proper estimation of pressure from this equation. Definitely, the
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more accurate the interpolated starred density (p ) values at the control volume faces are, the more accurate the predicted pressure values will be. The use of a central difference scheme for the interpolation of p leads to instability at Mach numbers near or above 1 [KAR 86, DEM 93]. On the other hand the use of a first order upwind scheme lead to excess diffusion [KAR 86]. The obvious solution would be to interpolate for values of p at the control volume faces the same way interpolation for other dependent variables is carried. That is to employ the bounded HR family of schemes for which no problem-dependent factors are required. Adopting this strategy, the discretized form of the starred continuity equation becomes:
The same procedure is also adopted for calculating the density when computing the mass flow rate at a control volume face in the general conservation equation. 5. Results and discussion The validity of the above described solution procedure is demonstrated in this section by presenting solutions to the inviscid flow over a bump. The physical situation consists of a channel of width equal to the length of the circular arc bump and of total length equal to three lengths of the bump. Results are presented for three different types of flow (subsonic, transonic, and supersonic). For subsonic and transonic calculations, the thickness-to-chord ratio is 10% and for supersonic flow calculations it is 4%. In all flow regimes, predictions obtained over a relatively coarse grid using the SMART scheme for all variables including density are compared against results obtained over the same grid using the SMART scheme for all variables except density, for which the UPWIND scheme is used. Due to the unavailability of an exact solution to the problem, a solution using a dense grid is generated and treated as the most accurate solution against which coarse grid results are compared. 5.1 Subsonic flow over a circular arc bump With an inlet Mach number of 0.5, the inviscid flow in the channel is fully subsonic and symmetric across the middle of the bump. Isobars displayed in Fig. 2(a) reveal that the coarse grid solution obtained with the SMART scheme for all variables falls on top of the dense grid solution. The use of the upwind scheme for density however, lowers the overall solution accuracy. The same conclusion can be drawn when comparing the Mach number distribution along the lower and upper walls of the channel. As seen in Fig. 2(b), the coarse grid profile obtained using the SMART scheme for density is closer to the dense grid profile than the one predicted employing the upwind scheme for density. The difference in results between the coarse grid solutions is not large for this test case. This is expected since the flow is subsonic and variations in density are relatively small. Larger differences are anticipated in the transonic and supersonic regimes.
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\~s
Figure 2. Subsonic flow over a 10% circular bump; (a) isobars and (b) profiles along the walls. 5.2 Transonic flow over a circular arc bump Results for an inlet Mach number of 0.675 are displayed in Fig. 3. Figure 3(a) presents a comparison between the coarse grid and dense grid results. As shown, the use of the HR SMART scheme for density greatly improves the predictions. Isobars generated over a coarse grid (63x16 c.v.) using the SMART scheme for all variables are very close to the ones obtained with a dense grid (252x54 c.v.). This is in difference with coarse grid results obtained using the upwind scheme for density and the SMART scheme for all other variables, which noticeably deviate from the dense grid solution. This is further apparent in Fig. 3(b) where Mach number profiles along the lower and upper walls are compared. As shown, the most accurate coarse grid results are those obtained with the SMART scheme for all variables and the worst ones are achieved with the upwind scheme for all variables. The maximum Mach number along the lower wall (si.41), predicted with a dense grid, is in excellent agreement with published values [DEM 93]. By comparing course grid profiles along the lower wall, the all-SMART solution is about 11% more accurate than the solution obtained using SMART for all variables and upwind for density and 21% more accurate than the highly diffusive all-upwind solution. 5.3 Supersonic flow over a circular arc bump Computations are presented for an inlet Mach number value of 1.4. Mach number contours are compared in Fig. 4(a). As before, the course grid all-SMART results (58x18 c.v.), being closer to the dense grid results (158x78 c.v.), are more accurate than those obtained when using the upwind scheme for density. The Mach profiles along the lower and upper walls, depicted in Fig. 4(b), are in excellent
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agreement with published results [NI 82] and reveal good enhancement in accuracy when using the SMART scheme for evaluating interface density values. The use of the upwind scheme to compute density deteriorates the solution accuracy even though a HR scheme is used for other variables. The all-upwind results are highly diffusive.
Figure 3. Transonic flow over a 10% circular bump; (a) isobars using various schemes, and (b) profiles along the walls.
Figure 4. Supersonic /low over a 4% circular bump (Min-1.4); (a) Mach number contours using various schemes, (b) profiles along the walls.
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6. Concluding Remarks A new high-resolution pressure-based algorithm for the solution of fluid flow at all speeds was formulated. The new features in the algorithm are the use of a HR scheme in calculating the density values at the control volume faces and the use of the NVSF methodology for bounding the convection fluxes. Results obtained were very promising and revealed good enhancement in accuracy at high Mach number values when calculating interface density values using a High-Resolution scheme. 7. Acknowlegments The financial support provided by the European Office of Aerospace Research and Development (BOARD) (SPC-99-4003) is gratefully acknowledged. 8. Bibliography [DAR 94] Darwish, M.S. and Moukalled, F.," Normalized Variable and Space Formulation Methodology For High-Resolution Schemes," Numerical Heat Transfer, Part B, vol. 26, pp. 79-96, 1994. [DEM 93] Demirdzic, I., Lilek, Z., and Peric, M.,"A Collocated Finite Volume Method For Predicting Flows at All Speeds," International Journal for Numerical Methods in Fluids, vol. 16, pp. 1029-1050, 1993. [KAR 86] Karki, K.C.,"A Calculation Procedure for Viscous Flows at All Speeds in Complex Geometries," Ph.D. Thesis, University of Minnesota, June 1986. [LEO 87] Leonard, B.P./'Locally Modified Quick Scheme for Highly Convective 2-D and 3-D Flows," Taylor, C. and Morgan, K. (eds.), Numerical Methods in Laminar and Turbulent Flows, Pineridge Press, Swansea, U.K., vol. 15, pp. 35-47, 1987. [LEO 88] Leonard, B.P.,"Simple High-Accuracy Resolution Program for Convective Modelling of Discontinuities," International Journal for Numerical Methods in Engineering, vol. 8, pp. 1291-1318, 1988. [LIE 93] Lien, F.S. and Leschziner, M.A.,"A Pressure-Velocity Solution Strategy for Compressible Flow and Its Application to Shock/Boundary-Layer Interaction Using Second-Moment Turbulence Closure," Journal of Fluids Engineering, vol. 115, pp. 717-725, 1993. [MAR 94] Marchi, C.H. and Maliska, C.R.,"A Non-orthogonal Finite-Volume Methods for the Solution of All Speed Flows Using Co-Located Variables," Numerical Heat Transfer, Part B, vol. 26, pp. 293-311, 1994. [NI 82] Ni, R.H.,"A Multiple Grid Scheme for Solving the Euler Equation," AIAA Journal, vol. 20, pp. 1565-1571, 1982. [PAT 81] Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere, N.Y., 1981. [RHI86] Rhie, C.M.,"A Pressure Based Navier-Stokes Solver Using the Multigrid Method," AIAA paper 86-0207, 1986.
Preserving Vorticity in Finite-Volume Schemes
Philip Roe and Bill Morton Department of Aerospace Engineering The University of Michigan Ann Arbor, Michigan 48109-2140 USA Department of Mathematical Sciences University of Bath Bath BA2 7AY United Kingdom
ABSTRACT We discuss the fact that many otherwise accurate finite-volume schemes have a tendency to yield anomalous solutions in certain circumstances. These are strongly linked to the appearance of spurious vorticity. For a model problem, we show that certain finite volume methods in fact preserve vorticity. Although these are not new schemes, they are not currently fashionable. A possibility exists to modify them so that they are are of high-resolution and upwind with respect to acoustic waves. Key Words: Conservation laws, Euler equations, vorticity
1.
Introduction
Vorticity is a very important aspect of many fluid flows, especially in three dimensions, on account of the great times for which it persists in a highReynolds number flow following its initial creation. The velocities "induced" by vorticity are essential to the operation of many fluid devices,and the behaviour of those regions where vorticity is concentrated are the key to the generation of sound. For computational purposes, incompressible flows are often formulated in variables that include vorticity, and "vortex methods" that explicitly track
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concentrated vortex cores are also widely used. For compressible flow, vorticity is hardly less important, but it is rare for it to be included in the computational formulation. Instead, the stress is placed on conservation so as to ensure the correct capturing of shocks. The consequence is that vortical structures are often badly diffused by numerical dissipation and this is a serious impediment to the prediction of complcated viscous flows. We have recently begun [MOR 99] construction of finite-volume methods that exert control over vorticity. We make no attempt to solve the vorticity evolution equations themselves, but try to find schemes such that a correct evolution of the discrete vorticity is an exact consequence of the discrete conservation laws, just as at the continuum level. Another way of expressing this is to say that whatever form of numerical dissipation is employed to stabilise the propagation of waves should have no impact on the advection of vorticity. There is a further motivation, which is that at the level of the Euler equations many otherwise accurate methods, especially those that attempt to minimise numerical dissipation by adopting upwind strategies (so that any disturbance is damped by the smallest damped compatible with its propagation speed) notoriously produce nonphysical solutions (carbuncle phenomenon) under some special (but not uncommon) circumstances. It now appears that these solutions are in fact quite legitimate solutions of the Euler equations, and could only be excluded by some selection principle over and above that of entropy-satisfaction. On the basis of examples encountered to date, all of these undesirable solutions appear to feature much greater vorticity that the anticipated solution. A very simple example is given in Section 2. There is also the related phenomenon discovered by Quirk [QUI 94]; that the same schemes that give rise to carbuncles can be "destabilsed" by very small grid perturbations. In simple academic examples it is usually easy to detect that something is wrong with the solution, and to tinker with the code until it goes away, but in situations where the flow is anyway expected to be complex there may be no simple way to detect unwanted phenomena. There is therefore an urgent need to understand these anomalies and to find means of avoiding them. 2.
Nonunique Solutions of the Euler Equations
Consider the supersonic flow past a flat-nosed two-dimensional body (although the argument works just as well for a flat-nosed cylinder with axial symmetry). We expect the solution to feature a rather smooth detached bow shock, giving rise to an embedded subsonic region, from which the flow accellerates smoothly to regain supersonic speed. This is shown in the top half of Figure 1. In the bottom half is an alternative solution. Plane oblique shocks originate from a more-or-less arbitrary point on the line of symmetry. Behind them is a triangular region of stagnant fluid extending to the shoulder of the body. Outside of this region the flow behaves exactly as it would if flowing past a triangular wedge (or cone) of solid matter. The pressure, velocity, etc.
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Figure 1. Two possible Euler solutions for the the flow past a flat-faced body. in that region may be found from the usual compressible tables and formulae. Prandtl-Meyer rarefaction waves return the flow to its original direction and weaken the initial shocks. Such a flow is in fact possible to create experimentally, and is illustrated in Plate 272 of [VAN 82] which is a schlieren picture taken in a windtunnel. The unusual flow configuration is the result of a thin "splitter plate" placed along the axis AB. Presumably what happens is that the boundary layer on the plate creates vorticity that is then diffused outward. In the limit of very high Reynolds number the vorticity eventually becomes concentrated in the infinitesimal shear layers AC, AD There is no net circulation generated in the flow but each half of the stagnant region (above and below the axis) has circulation around it. This flow is a valid second solution of the Euler equations, provide that the pressure in the stagnant region matches the pressure outside it. The temperature in the stagnant region is arbitrary if we allow that an entropy layer might also exist, so there is a doubly-infinite family of solutions. All the shocks are entropy-satisfying, but under some conditions the shear layers might not be stable. One may now imagine that the corners of the body are steadily rounded off until the nose becomes a semi-circle. The alternative solutions would become the carbuncles. According to Pandolfi and D'Ambrosio, who have made very detailed observations [PAN 99] the solutions produced in the carbuncle phenomenon are also genuine Euler solutions. One might hope that by solving the Navier-Stokes equations instead the carbuncle would automatically disappear. However, Gressier and Moschetta
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[GRE 99] report that the Reynolds number had to be reduced to as little as 100 before this happened. It seems that the mechanisms by which the Euler code introduces the vorticity is stronger than the mechanism by which the Navier-Stokes code removes it. The Quirk phenomenon occurs when a plane one-dimensional shock is propagated along one direction of a rectangular grid. If the grid is indeed perfectly rectangular then one recovers the one-dimensional solution but if one row of grid points, parallel to the direction of motion, is very slightly perturbed, an exponentially-growing pattern of high-frequency disturbances is generated that eventually yields a flow pattern not unlike that seen in carbuncles. Robinet et al [ROB 99] attempt to relate this to a new observation concerning the linear stabilty of a plane shock. Although this is a classical flow stability problem, they have discovered a 'strange mode' overlooked by previous analysts. They find perturbations of a plane two-dimensional flow by separation of variables. This leads to an eigenvalue problem having linearly independent solutions except for certain combinations of frequency in time and wavenumber in space. Rather unexpectedly, these exceptional solutions involve a resonance in which an acoustic mode and a vortical mode become indistinguishable. The linear algebra problem has to be completed by a Jordan block with a generalised eigenvector. Coupling this with the equations governing the shock perturbation reveals a solution that can grow exponentially with time. Many features of this solution are shown to appear in numerical experiments on the Quirk phenomenon. Experience to date is that all numerical schemes that display the carbuncle phenomenon also display the Quirk phenomenon. There is a proof [GRE 98] that all schemes capable of resolving a parallel shear layer without dissipation (a highly desirable property, and one responsible for the widespread adoption of flux-difference-splitting schemes rather than flux-splitting schemes) will in fact display the Quirk phenomenon. This collection of facts almost suggests a crisis situation. In the next section we present analysis, condensed from [MOR 99] that opens an avenue of escape. 3.
A Model Problem
The simplest model problem to combine wave propagation and vorticity is the system wave equation. We will write this in two space dimensions in the matrix form, using a notation corresponding to acoustic waves in a fluid that is stationary in the mean, with pressure p* and velocity u* = (u*,v*), thus dtu+cLu = Q.
(1)
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Figure 2. Grid definition. Here u = (p*/(pc2), u*/c, v*/c) = ( p , u , v , ) , c is the sound speed in the mean flow, and
Restriction to two dimensions is merely for economy of notation; all of the analysis extends very straightforwardly to three dimensions. We study the wave equation in system rather than scalar (dttu — c 2 V 2 u) form for two reasons. Firstly because this is the form of the wave equation that is hidden inside the Euler equations, whether in their two-dimensional unsteady or in their three-dimensional supersonic steady forms. Secondly, because the scalar form automatically implies vanishing vorticity, whereas the interaction of the waves with vorticity is one of the aspects we want to study. Here the interaction is very simple, as befits a model problem. We easily deduce from (1,2) that dt£ = 0,
where
£ — dxv — dyu.
In other words, there is no interaction and any initial distribution of vorticity is preserved. Maintaining this independence at the discrete level will be our objective. 3.1.
Discrete Notation
In this paper we concentrate on simple finite-difference formulations on uniform square grids, such that the spacing in the x and y directions is h and the time step is At, with ufj a discrete approximation located at ( x , y , t ) —
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(ih,jh,nAt). The standard discrete differencing and averaging operators are defined by
The points with integer coordinates will be called cells, those with one integer and one half-integer coordinate edges and those with two half-integer coordinates vertices. The variables stored at cells will be (p,«,?;). The same variables stored at vertices will be distinguished by primes where neccessary. The variables stored at vertical edges will be (P, U) and those stored at horizontal edges (Q, V). In a finite-volume interpretation the edge quantities are the fluxes. (See Figure 2) If it is recognised that each application of the above operators moves the mesh values to a different set of grid points then arbitrarily long products of operators are allowed and all multiplications commute. 3.2.
Conservation Form
In the cell-centred finite-volume method, discrete conservation is ensured by drawing a control volume around the grid point of interest, and writing the update as an integral around this volume. In the generic case of a vector U of conserved variables, with fluxes F, G in the (x, y)-directions respectively, one has
where F*, G* are numerical fluxes evaluated from some formula to be determined. In the present case we can write, with v = c&i/h and following the notation of Figure 2,
It will usefully restrict the schemes to require that they can be written in this form. A second-order scheme of the Lax-Wendroff type follows from taking U, V, P, Q to be estimates halfway through the time step. However, we will find subsequently that it is a rather special type of conservation form that emerges from the analysis.
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Preserving Vorticity
The first step in creating a scheme that preserves vorticity is to define a discrete vorticity, which we do as follows
This will be preserved if
and the condition ^ixP = ^yQ will be met if we take P = nyr' ,Q = nxr' where r' is some quantity defined at vertices. The only way to define a consistent local pressure while retaining a nine-point stencil is to take
In that case we have
To obtain second-order accuracy, r' must now be updated to half-way through the time step. The simple formula
is the unique symmetrical formula to achieve this without enlarging the stencil, leading to
We remark here that a general example of the Lax- Wendroff family will generate vorticity at a rate proportional to h3; this is shown in [MOR 99].
3.4-
Complete Evolution Operator
We construct the matrix operator that will update the solution, so that if
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certain elements of the matrix MA are already uniquely determined by insisting that the velocities are updated with second-order accuracy while preserving the discrete vorticity (6). We have in fact, from (4),(5),(9),(10),
The adjoint property div/j = —grad£ requires that this matrix be symmetric [MOR 99] and an agument based on maintaining a compact stencil leads to the complete matrix operator
The scheme represented by this matrix has been uniquely determined by the requirements of conservation, vorticity preservation, symmetry of the solution under grid transformations, adjoint symmetry of the discrete operator, and second-order accuracy. However it is not a new scheme. It can be recognised by noting that MA can be factored as
where
and therefore can be written as a two-step scheme. The operation
gives a provisional solution at the vertices. The operation
completes the update by integrating around the vertices. This is in fact the version of Lax-Wendroff known as the Rotated Richtmyer scheme[RIC 62]. It is shown schematically on the left of Figure 3. The original motivations for this scheme were compactness, computational economy and stability. In the nonlinear case, as in all two-step Lax-Wendroff schemes, one avoids any multiplication by the Jacobian matrices. The vorticity-preserving property does not seem to have been previously noticed. It is shown in [MOR 99] that the scheme is stable for the maximum possible CFL range cAt/h < I.
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Figure 3. (left) The Rotated Richtmyer scheme. In the first step, symbolised by white arrows, data from the cells is used to create a half time-step solution at the vertices. In the second step (black arrows) integration round the vertices updates the central cell, (right) Ni's cell-vertex scheme. In the first step (white arrows) we integrate around the cells to obtain a 'cell-residual'. In the second step (black arrows) these are distributed to the vertices. 3.5.
Duality
Since both factors of MA depend only on LA they commute, and so the scheme may also be written as
In this form it is Ni's cell-vertex scheme [NI 82], in which the variables are usually thought of as located at vertices of a grid, defining a bilinear interpolant over a square element. The first step LA is to integrate dxF + dyG over this element and the second step is to distribute this to the nodes of the element. This distribution operation is described by the first factor in (17) 4.
Commentary
By choosing a correct definition of discrete vorticity, the results above can be extended [MOR 99] to the linear wave equation on unstructured two- and three-dimensional meshes, and to problems with non-constant coefficients for which vorticity should be created. In the latter case there is a discrete Kelvin Theorem giving the growth of circulation around a certain class of contour. All of this is of course only a beginning to the design of practical schemes, but an important observation is that the vorticity-preserving property does not depend on choosing any particular expression for the quantity r' in (7). It is enough to compute as a first step any vertex pressure, even a ridiculous one, and then to find the edge presure by averaging along the edges. This gives scope for introducing nonlinear methods (limiter functions) to control the os-
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dilations around Shockwaves without introducing vorticity. It is beyond the scope of this paper to discuss details, but it is possible to design schemes that preserve vorticity in two or three dimensions, and which reduce in one dimension to high-resolution upwind schemes. However, the two-dimensional fluxes are computed from six rather than two neighbouring states, which takes the schemes outside the applicability of the results in [GRE 98]. Therefore schemes that preserve contact discontinuities but avoid carbuncle-type behaviour may still be possible. We are currently investigating such schemes. For the fully nonlinear Euler equations, the evolution of discrete vorticity is quite complicated. However, the evolution of the curl of the momentum is V x (pv) is rather simple, and to avoid unwanted solutions it may be enough to control this quantity. We hope to be able to report shortly on the outcome of numerical experiments. 5.
Bibliography
[MOR 99]
MORTON, K. W.. AND ROE, P., "Vorticity-preserving Schemes of LaxWendroff Type", Submitted to SI AM J. Sci. Comp., 1999.
[ROB 99]
ROBINET, J-CH., GRESSIER, J., CASALIS, G. AND MOSCHETTA, J-M., "Shock Wave Instability and Carbuncle Phenomenon- Same Intrinsic Origin?", preprint, ONERA Toulouse, submitted to J. Fluid Mech., 1999.
[QUI 94]
QUIRK, J. J., "A Contribution to the Great Riemann Solver Debate", Int. J. Num. Meth. in Fluids, 18, pp. 555-574, 1994
[VAN 82]
VAN DYKE, M., An Album of Fluid Motion, Parabolic Press, Stanford, CA., 1982
[PAN 99]
PANDOLFI, M. AND D'AMBROSIO, D., "Numerical Instabilities in Upwind Methods: Analysis and Cures for the "Carbuncle" Phenomenon", preprint, Politechnico di Torino, submitted to J. Comput. Phys., 1999
[GRE 99]
GRESSIER, J. AND MOSCHETTA, J-M., "Robustness versus Accuracy in Shock-Wave Calculations", preprint, ONERA Toulouse, submitted to Int. J. Num. Meth. in Fluids, 1999.
[GRE 98]
GRESSIER, J. AND MOSCHETTA, J-M., "On the Pathological Behaviour of Upwind Schemes", AIAA Paper 98-0110.
[RIC 62]
RICHTMYER, R. D.,"A survey of difference methods for non-steady fluid dynamics", NCAR Tech. Note 63-2, Nat'l. Center for Atmos. Research, Boulder, CO, 1962.
[NI 82]
Nl, R-H.,"A multiple-grid scheme for solving the Euler equations", AIAA Jnl, 20, p 1565, 1982.
On Uniformly Accurate Upwinding for Hyperbolic Systems with Relaxation
Jeffrey Hittinger and Philip Roe Department of Aerospace Engineering The University of Michigan Ann Arbor, Michigan, USA 48109-2140
ABSTRACT The design of uniformly accurate, upwind Godunov schemes for hyperbolic systems with relaxation source terms is discussed. The goal is to develop upwind methods whose sole time step constraints are due to the advection terms yet which obtain accurate solutions even when the relaxation terms are underresolved. An archetypal model system is considered, and analysis of the Riemann initial value problem for this model system is discussed. The ideas learned from this are re-enforced by an asymptotic analysis of general Riemann problems, and these results are described. A strategy for developing a suitable numerical flux function is then outlined. Key Words: hyperbolic systems, relaxation, numerical upwinding, stiff source terms
1.
Introduction
Many flow problems such as those of dilute gases or fluid mixtures have natural formulations as hyperbolic systems with relaxation source terms. These pose interesting challenges for numerical approximation. The source terms cause the system to be dispersive; generally, both the eigenvalues and eigenvectors of the system are modified by the relaxation processes which drive the system towards equilibrium. Another issue arises when the relaxation source terms operate on much smaller time scales than the advection terms; the problem is then said to be stiff. Unfortunately, many problems simultaneously exhibit behavior in both the stiff and non-stiff limits, as well as in between. It is therefore desirable to develop high-resolution numerical algorithms which are uniformly accurate at all scales. For example, if the data are such that
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the advection terms permit a time step large enough to bring the flow into local equilibrium, one would wish to see an accurate solution of the equilibrium problem emerge without resolving the details of the transition. In recent years, there has been much research into such uniformly accurate methods, although a clearly satisfactory procedure has yet to emerge. Pember's early work in the area [PEM 93] identified many properties desirable for Godunov-type schemes for hyperbolic systems with stiff relaxation but did not resolve the relative significance of the frozen and equilibrium wave speeds. Splitting schemes [CAP 97, JIN 96], have been proposed, but general objections to splitting were made in [ARO 96, ROE 93]. These authors proposed characteristic schemes but found no natural way to make them conservative. A conservative method based on an approximate linearization was proposed in [BER 97]. This uses a staggered grid to eliminate the difficulty of resolving the Riemann problem, which no longer has a self-similar solution. However, such schemes may lose definition of linearly degenerate waves, such as contact discontinuities, which are likely to be very important in reactive or relaxing flows. Currently we are attempting to develop an approximate Riemann solver that represents the physics with uniform accuracy and whose computational costs are acceptable. The general behavior of relaxing Riemann problems is now known thanks to the work of Liu and Zeng [ZEN99], although they have concentrated mostly on qualitative behavior of rather general systems. It is complementary to our own work which has involved very detailed analysis of a linear model system which we feel is archetypal. This model has an exact (integral) solution for the Riemann initial value problem, which allows one to see how a simplified solution might be developed.
2.
A Model System
To gain a better understanding of the behavior of hyperbolic systems with relaxation source terms, we have studied the linear1 system
where x € M. and t € R+ are the spatial and temporal independent variables, respectively; e £ K+ is a constant relaxation time; and r e [0,1] corresponds to the equilibrium wave speed. 1 Only the linear problem is considered for the obvious reason that analysis is made tractable. However, this is not the restriction it might seem, as other authors [BER 97] have identified a suitable linearization for nonlinear hyperbolic systems with relaxation. We will demonstrate their requirements in Section 3.2.
The first equation (la) describes the evolution of the conserved quantity u(x,t). The flux of u, that is v(x,t), has its own evolution equation (Ib), and is driven towards equilibrium over a time scale O(e) by the relaxation source term. In equilibrium, this source term vanishes (v = ru), and the system reduces to the single linear advection equation
with the equilibrium wave speed r. Near equilibrium, say v — ru + O(e), it can be shown that
which is an advection-diffusion equation. In similar analysis applied to more complex problems, such as moment approximations of the Boltzmann equations, (2) corresponds to the Euler equations, and (3) to the Navier-Stokes equations. 3. 3.1.
The Riemann Problem The Model Problem
For upwind schemes, the Riemann problem is the heart of the numerical algorithm. We will describe the Riemann problem for the model system and point out behavior that is generic to all relaxation systems. Setting (f)x = —u and (j>t = v allows the system (1) to be reduced to a scalar equation, to which an exact integral solution can be constructed using classical methods. (For details see [ROE 93].) This integral solution is amenable to asymptotic analysis, particularly at small times, and can also be evaluated numerically to provide a complete picture of the evolution of the solution from the small-time to the long-time asymptotics. Consider the system (1) in vector form
with the piecewise constant initial conditions
The discontinuity in value at the origin has a domain of influence bounded by the frozen characteristics |£| = \x/t\ = 1. Outside of the domain of influence, the initial conditions are constant, and, hence, the system of partial differential equations (4) reduces to a system of ordinary differential equations in time. This system is easily integrated to find
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where r — t/e. The conserved variable u remains constant while the relaxation variable v decays exponentially from its initial value to the equilibrium value ru. This rapid adjustment of the relaxation variable forms a temporal boundary layer, as is apparent in the contour plots presented in Figure 1. Within the domain of influence, the solution is
where
with ry* = t — x, (* = t + x, T^o — 7?.(0,0), and the Riemann function
is the product of an exponential and the modified Bessel function of the first kind IQ. Examination of this solution shows that, just behind the frozen waves, the solution decays exponentially to the equilibrium state ahead of each wave. That is, the strengths of the frozen waves are decaying exponentially, as can be seen in Figure 1. The actual decay rates are exp(—(1 ± r)r/2). For small times, r = t/e —> 0 , the solution for u(x,t) is a perturbation about the frozen solution UQ, with the O(n) corrections each a polynomial of degree n in the variable £:
where
with
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Figure 1. Contour plots of the exact solution for the Riemann problem U£ = (1,5)T and UR = (10,3)T for the equilibrium wave speed of r — 0.5 and relaxation time e — 0.01. Left-hand plots show a small-time behavior while the right-hand plots demonstrate a longer-time behavior. The dotted lines \x/t\ — 1, representing the frozen waves, are drawn to emphasize the decay of these waves.
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The piecewise-constant frozen solution is replaced by a smoothly varying solution as the result of decay along and interaction between the characteristics. The diagonalization of (4) explicates this behavior of the characteristics:
Here, v = ~(u + v,u — v)T is the state vector of the frozen characteristic variables. On the right-hand side, the diagonal elements correspond to the previously-noted decay along the characteristics. The off-diagonal elements characterize the interaction between characteristics. For long times, T —>• oo, between the original two waves, the integral factors in TL and TR can be expanded by Laplace's method. The leading-order terms of the solution are
In this limit, the original two waves vanish and are replaced by the equilibrium wave at £ = r. In Figure 1, the rise of the equilibrium wave can easily be seen. As the small-time asymptotic behavior decays, the solution behind the frozen waves tends to the states ahead of the waves except in the narrow region of the equilibrium wave. Here, the solution varies rapidly but smoothly from the left state to the right state. Finally, we consider the flux at £ = 0, which corresponds to a cell interface in an upwind Godunov scheme. For the same Riemann problem as Figure 1, the interface state is plotted in Figure 2 together with the small-time and large-time asymptotic expansions. Note that the interface flux is just (v,u)T'. The flux begins as the frozen flux, then varies smoothly but not necessarily monotonically to the equilibrium flux with this transition in the vicinity of r — 1, where the relaxation and advection are equally important.
3.2.
General Linear Systems
Consider the general linear system in the form (4), where now u G Mm and A,Q € E m x m . We assume that rankQ = (m - n) > 0, so that Q has a null space AA(Q), which is the equilibrium manifold in which the equilibrium solution takes place. Let L0 G M nxm be the row matrix of the left eigenvectors of Q which span AT(Q); similarly, let RQ G R m x n be the column matrix of the right eigenvectors of Q which span Af(Q). It is easy to show that the equilibrium (Euler) equation is
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Figure 2. The. interface state for the Riemann problem UL = (1,5) T and UK — (10,3) T for the equilibrium wave speed of r — 0.5 and relaxation time c = 0.01. The solid line is the numerical calculation of the exact solution, the dotted line is the small-time asymptotic expansion up to terms O(r 4 ), and the dashed line is the leading-order long-time asymptotic solution. and the late-time solution to the Riemann problem comes from the self-similar solution to this problem with Riemann data L 0 U£,,LoUR. In [BER 97], it was proposed to select an average matrix Q(U£,U#) such that Lo(Q)ARo(Q) is the correct Roe matrix for the equilibrium problem. Because they avoided consideration of the early time Riemann problem, they had no need to select any particular linearization of A, but if this is simply a standard Roe linearization, then isolated discontinuities will be correctly captured in both limits. If n — (m — 1), the solution is only stable if the equilibrium wave speeds (the eigenvalues of LoARo) precisely interleave the eigenvalues of A [LIU87]. For this case, an equilibrium wave will grow from the smooth flow between each pair of frozen waves, and simultaneously, this equilibrium wave will be diffused. The frozen waves decay unless they carry a jump lying in A^(Q); such waves will be found in both the early and late time solutions. If n < (m — 1), are no theoretical results, but the general picture of frozen shocks undergoing exponential decay with equilibrium shocks arising from the spaces between them appears to be correct. At early times one can obtain expansions similar to (12). The fc-th frozen wave decays such that
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where \k and r^ are the fc-th left and right eigenvectors of the matrix A, and ct;t(0) is the strength of the initial jump. The solution between the waves is again piece wise polynomial of degree n at order n.
4.
Constructing an Upwind Flux Function
The substitution u = exp{Qr}w transforms (4) into wt + exp{-Qr}Aexp{Qr}wx = 0.
(21)
Integrating this around a cell of a regular, uniform grid with cell-centered data, and translating back to the original variables, one finds
where t1 = t — nAt and f = Au. If the solutions fj±i(t) are known on the interface, this is an explicit formula for the new cell average. The exponential decay in the transient solution is represented explicitly, and within the integral, the exponential factor acts as a filter. To elucidate the weighting in the integrand, let z = Lu, where L is the row matrix of left eigenvectors of Q. This transformation creates a state vector of the n conserved variables (those in JV(Q)) and of the (m — n) relaxation variables which vanish in the equilibrium limit. Multiplying the update formula by L, one obtains
where f = LAL lz and A is the diagonal matrix of the eigenvalues of Q. For conserved variables, the corresponding eigenvalues are zero, so the exponential factors in the update (23) are just unity. For relaxation variables, interpreting At as the time step based solely upon the advection terms, the exponential weighting in the integrand is essentially unity over the entire interval if At/e
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The difficulty is only with those cells for which Ai/e = 0(1). An empirical blending of the two formulae will probably not be very accurate, because of the non-monotone transition seen in Figure 2. However, Figure 2 also shows that the early-time expansion may have a sufficiently large radius of convergence that it can capture the crucial behavior. We are presently investigating such expansions in the general case. The solution for the jumps is given above, and the solution for the linear variation is also straightforward. The quadratic and higher terms appear more complicated, although there is a straightforward means to derive them. But even if the formulae prove expensive they should only be needed at relatively few interfaces. 5.
A Practical Example
Between the continuum (Euler) model of fluid flow and the molecular (Boltzmann) description, there lie many intermediate models. The hierarchy of models devised by Levermore [LEV 96] are of especial interest, because by design, they possess many properties required for trouble-free computation: they are hyperbolic, symmetrizable, entropic, and well-posed. Beyond the Euler equations, the next member of the hierarchy retains ten moments of the Boltzmann equation by assuming that the distribution of the random velocity c is proportional to exp(—^©"^CjCj), where QIJ is a symmetric, non-negative 3x3 matrix closely related to the temperature. Moments of this distribution give the pressure and temperature as tensor quantities. The relaxation process is simply that the temperature tries to relax back toward a scalar,
and the eigenstructure of the frozen problem is straightforward (See [BRO 95]). We feel that although the procedures outlined in this paper may be expensive in general, they will prove relatively simple for well-motivated physical models. 6.
Conclusions and Future Work
The analysis of a simple linear model system has provided clues towards the development of a uniformly accurate upwind method for hyperbolic systems with relaxation. Specifically, the solution of the Riemann problem can be constructed and analyzed for this model system, and this analysis has identified strategies for designing upwind methods for hyperbolic systems with relaxation source terms. Currently, we are implementing and evaluating these ideas.
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Acknowledgment
This work is supported in part by a U.S. Department of Energy Computational Science Graduate Fellowship.
8.
Bibliography
[ARO 96]
ARORA, M., Explicit Characteristic-Based High-Resolution Algorithms for Hyperbolic Conservation Laws with Stiff Source Terms, Ph.D. thesis, The University of Michigan, 1996.
[ARO 98]
ARORA, M. AND ROE, P., "Issues and Strategies for Hyperbolic Problems with Stiff Source Terms", Barriers and Challenges in Computational Fluid Dynamics, V. Venkatakrishnan et al., eds., Kluwer Academic Publishers, Norwell, MA, 1998, pp. 139-154.
[BER 97]
BEREUX, F. AND SAINSAULIEU, L., "A Roe-type Riemann Solver for Hyperbolic Systems with Relaxation Based on Time-Dependent Wave Decomposition", Numer. Math., 77, 2, 1997, pp. 143-185.
[BRO 95]
BROWN, S., ROE, P., AND GROTH, C., "Numerical Solution of 10Moment Model for Nonequilibrium Gasdynamics", AIAA Paper 95-1677, June, 1995.
[CAP 97]
CAFLISCH, R., JIN, S., AND Russo, G., "Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation", SIAM J. Numer. Anal., 34, 1, 1997, pp. 246-281.
[JIN 95]
JIN, S., "Runge-Kutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms", J. Comput. Phys., 122, 1, 1995, pp. 51-67.
[JIN 96]
JIN, S. AND LEVERMORE, C., "Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms", J. Comput. Phys., 126, 2, 1996, pp. 449-467.
[LEV 96]
LEVERMORE, C., "Moment Closure Hierarchies for Kinetic Theories", J. Stat. Phys., 83, 5/6, 1996, 1021-1065.
[LIU87]
T.-P. Liu, " Hyperbolic conservation laws with relaxation", Math. Phys, 108, 1, 1987, pp. 153-175.
[PEM 93]
PEMBER, R., "Numerical Methods for Hyperbolic Conservation Laws with Stiff Relaxation: II. Higher-Order Godunov Methods", SIAM J. Sci. Comput, 14, 4, 1993, pp. 824-859.
[ROE 93]
ROE, P. AND ARORA, M., "Characteristic-Based Schemes for Dispersive Waves: I. The Method of Characteristics for Smooth Solutions", Numer. Methods Partial Differential Equations, 9, 5, 1993, pp. 459-505.
[ZEN99]
ZENG, Y., "Thermal Nonequilibrium and General Hyperbolic Systems with Relaxation", preprint, University of Alabama at Birmingham, 1999.
Comm.
Implicit Finite Volume approximation of incompressible multi-phase flows using an original One Cell Local Multigrid method
Stephane VINCENT and Jean-Paul Avenue Pey-Berland BP 108 33402 Talence Cedex France
CALTAGIRONE
ABSTRACT The numerical simulation of multi-phase flows involving stretching and tearing of interfaces requires accurate tools, able to describe near the free surface the different scales of the flow which results from the development of instabilities. On fixed Cartesian mesh, an original local multigrid method, which refines the grid at the cell scale and adapts in time and space, is proposed. An implicit Finite Volume solver, coupled with a TVD- VOF like interface capturing method, is carried out on each grid level. The method is validated and discussed on analytical velocity fields and Rayleigh-Taylor instabilities. Key Words: free surface flows, multigrid method, implicit Finite Volumes
1. Introduction
The numerical simulation of multi-phase flows with strong stresses acting on the interface is classically achieved by the implementation of fixed Cartesian meshes with an interface tracking method (Marker [DAL 67], VOF [YOU 82] or Level Set [SUS 97]). However, due to the memory limit of supercomputers and the computational time, the numerical simulation of non-symmetric threedimensional free surface flows is restricted to problems where the length-scales of the phenomena occurring near the interface are close. To limit the computation node far away from the free surface and concentrate the calculation points on the interface, an original One Cell Local Multigrid method (OCLM) is proposed. Starting on a coarse grid GO which corresponds to the physical domain 17, a refinement criterion Rc is defined to detect the points to be refined on
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each multigrid levels GI, 1 < / < lmax- A hierarchy of embedded subdomains GI,S, 0 < s < smax, is obtained, each fine calculation grid GI>S corresponding to a coarse control volume around a detected point of G/_i cut by 3 in each space direction. An odd cutting ensures a perfect reconnection between the solutions on each GI (Fig. 1). Contrary to the classical multigrid methods such as FIC [ANG 92] or AMR [BER 89], where the fine calculation meshes contains tens to hundreds of cells, all the multigrid calculation domains have the same size (3 x 3) in the OCLM technique, which is an outstanding property of the method. Indeed, on meshes of reduced size, the numerical solvers converges quickly and requires very low memory.
Figure 1. Local mesh refinement technique at the scale of a control volume around a coarse node. After a brief presentation of the motion equations, the numerical solver on a unique grid is explained. The various stages of the OCLM method are next described and finally, the local refinement technique is discussed on classical interfacial problems. 2. 1-fluid model and unique grid solver
The multi-phase flow is modeled by means of the dimensional Navier-Stokes equations for incompressible fluids. In Cartesian coordinates, with a bounded domain fi, we get
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where p is the pressure, u is the velocity field, C is the phase function , g is the gravity, p is the density and p, is the dynamic viscosity. The colour function C repairs the different phases of the flow standing for example C = 0 in one fluid and C — 1 in the other. The interface between these fluids is naturally defined as C = 0.5. The motion equation system (1-3) is a 1-fluid model, in which the discontinuous physical characteristics are estimated according to the arrangement of the fluids as follows
where /?o,pi,/^o and /^i are the physical characteristics of fluids 0 and 1. Finite Volumes on a staggered grid (MAC) are investigated to discretise the Navier-Stokes equation system (1-3). In (1-2), the temporal derivatives are approximated by a Gear scheme of second order, whereas the discretisation of the spatial derivatives is achieved through a Quick scheme for the non-linear terms and a centred scheme for the diffusive one. Moreover, in the presence of discontinuous physical characteristics, a robust augmented Lagrangian numerical solver (Vincent and Caltagirone [VIN 99]) is carried out to calculate the solution of (1-2). The coupling between the pressure p and the velocity u is gone around thanks to a penalization terms added in the momentum conservation equation (Fortin and Glowinski [FOR 82]). Then, the implicit discrete equation system reads
where At is the time scale, r is a numerical parameter controlling incompressibility and n is a normal to F. The exponent n corresponds to time (n At) and subscript 0 refers to the coarse grid GQ. u^ is a reference velocity and B^ is a volume control parameter, which is used to impose a velocity in fi (Angot [ANG 89]) whereas B£ is a Fourier like surface control parameter enforcing boundary conditions on F. An iterative Bi-Conjugate Gradient Stabilised algorithm (BiCgStab, Van Der Vorst [VAN 92]), preconditionned with a Modified Incomplete LU method (MILU) is investigated to solve the linear system gen-
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erated by the discretisation of the motion equations. In the presence of a discontinuous phase function C, the hyperbolic advection equation (3) must be discretised by means of non-oscillating TVD schemes of high order (Vincent and Caltagirone [VIN 99]). In this way, no spurious oscillations appear and a second order accuracy is obtained in each fluid. At the interface, the order of the TVD scheme used is decreased to ensure the monotonicity of the solution. With the implementation of a Lax-Wendroff TVD scheme, accurate solutions were obtained on complex multi-phase flows (see [VIN 99] and [VIN2 99]). 3. One Cell Local Multigrid solver
If it is supposed that the solution (UQ ,PQ , CQ] of (5) on the coarse grid GO at time (n At) is known, then (UQ+I ,p£+l ,CQ+I) can be calculated solving (5) on GO- A refinement criterion Rc is defined to detect the coarse points where a multi-scale solution is necessary. For free surface flows, a local mesh refinement is built on the coarse control volumes cut by the interface. In this way, Rc is expressed as If Rc (M) ^ 0 with M € GI-I, the control volume around M is refined and a fine calculation domain G^s is created. For all I such that I
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Dirichlet boundary condition enforces the numerical solver to converge to a non suitable solution. To relax the stresses at the boundaries of the multigrid calculation grids, composite boundary conditions have been introduced in the Navier-Stokes equation system as follows
A theoretical work was developed to impose the conservation of the shear stress between two multigrid levels (see [VIN3 99]). In this way, orders of magnitude of the composite boundary condition parameters B^ and B^ were estimated, included between 104 and 109. Moreover, the Navier-Stokes solving on the 5 x 5 fine grids is improved thanks to an incomplete ILUD preconditionnning, more powerful than the MILU one.
4. Results
4.1. The vortex test A concentration circle of radius 0.15 meter located at point (0.5, 0.75) in a 1 meter-long square domain is stretched in the symmetrical vortex velocity field of Rider and Kothe [RID 95]. The authors demonstrated that only the Marker method can reproduce the fine scale of the solution. However, this technique is very expensive. All the other methods tested in their paper destroy the fine features of the interface. The Level Set technique and the TVD scheme solution induce strong diffusion and lose precision when the interface is strongly stretched. The PLIC VOF method introduces artificial surface tension in the small scale regions of the interface which are artificially cut. To be efficient, these techniques require very fine calculation grids. Thanks to the OCLM method, coupled with a TVD interface capturing method, the problem is precisely solved with a very low number of calculation points (Fig. 2). Starting with a 70 x 70 coarse grid and applying the OCLM method on two levels, a precision equivalent to 630 x 630 is reached in interfacial regions. Thanks to the local mesh refinement, the memory costs have been divided by 4 with respect to a computation on a full 630 x 630 grid. The local and adaptative character of the OCLM method is perfectly illustrated in this problem.
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Figure 2. Multigrid simulation of the vortex problem. G0 is a 70 x 70 grid. A 3-level solution is presented after large deformations have been induced. The local mesh refinement (only 40% of the total cells are plotted for convenience) and the interface position (C=0.5) on G^ are presented.
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4.2. Rayleigh-Taylor instabilities
Figure 3. Numerical simulation of two-dimensional Rayleigh-Taylor instability by means of the OCLM method. The initial perturbation is 10% of the domain height. The viscosities are the same in the two fluids. The results are presented respectively at time 2.25 s. The distribution of the multigrid cells on the coarse grid and the free surface (C = 0.5) are plotted. The Atwood number A is 0.05 and the Reynolds number Re is 10. The Rayleigh-Taylor instability is a classical and widely studied phenomenon which underlines the competition between the surface tension forces and the viscous stresses. When a heavy fluid lays above a lighter one, any perturbation of the interface between these fluids is amplified under the action of gravity,
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whereas the surface tension tends to minimise the deformation of the free surface. The Rayleigh-Taylor instability problem is simulated to highlight the abilities of the OCLM method to solved free surface flows. The Navier-Stokes equations and the interface capturing procedure are computed on all the grid levels. For example, the local mesh refinement is carried out on 2 grid levels (Fig. 3). GO is a 40 x 80 grid. The free surface evolutions are perfectly captured by the OCLM method. The local character (4 or 5 cells surrounding the interface) and the adaptative property of the method in time and space are verified. The composite boundary conditions have allowed to solved precisely the motion equations on the fine calculation domains. Moreover, the reconnection between the hundreds of fine grids is perfectly achieved thanks to the odd cutting. In the present problem, the ratio between the number of calculation point in the OCLM solving and the number of node on an equivalent fine 120 x 240 grid is always less than 0.09. The gain in memory is so very important and the calculation time is decreased about 10 %. The difference between the memory and calculation time improvements lays in the additional operations needed to solved on the multigrid levels and in the detection procedure.
5. Conclusion
We have presented a new local and adaptative multigrid refinement method (OCLM) for the numerical simulation of free surface flows. The motion equations and the interface capturing are solved at the cell scale thanks to the definition of new composite boundary conditions. The method brings important gains in memory and calculation time. The extension of the method to three dimensions is immediate. Moreover, the OCLM method is a general method independent on the equations solved as well as on the numerical solver. The method was successfully applied to the numerical simulation of natural convection in a square cavity. [DAL 67]
DALY B. J., Numerical study of two-fluid Rayleigh-Taylor instability , Phys. Fluids, 10, 1967, p. 297-307.
[YOU 82]
YOUNGS D. L., Time-dependent multi-material flow with large fluid distortion, K. M. Morton and M. J. Baines, 1982, p. 27-39.
[SUS 97]
SUSSMAN M. AND SMEREKA P., Axisymmetric free boundary problems , J. Fluid Mech., 341, 1997, p. 269-294.
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[ANG 92]
ANGOT P., CALTAGIRONE J.-P. AND KHADRA K., Une methods adaptative de raffinement local: la correction du flux a 1'interface , C. R. Acad. Sci. Serie II b, 315, 1992, p. 739-745.
[BER 89]
BERGER M. J. AND COLLELA P., Local adaptative mesh refinement for hyperbolic partial differential equations , J. Comput. Phys., 82, 1989, p. 64-84.
[VIN 99]
VINCENT S. AND CALTAGIRONE J.-P., Efficient solving method for unsteady incompressible interfacial flow problems , Int. J. Numer. Methods Fluids, 1999, to be published.
[FOR 82]
FORTIN M. AND GLOWINSKY R., Methodes de Lagrangien augmente. Application a la resolution numerique de problemes aux limites, Dunod, 1982.
[ANG 89]
ANGOT P., Contribution a 1'etude des transferts thermiques dans des systemes complexes: application aux composants electroniques, PhD thesis, University Bordeaux I, 1989.
[VAN 92]
VAN DER VORST H. A., Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems , SIAM J. Sci. Stat. Comput., 13, 1992, p. 631644.
[VIN2 99]
VINCENT S., CALTAGIRONE J.-P. AND ARQUIS E., Numerical simulation of liquid metal particules impacting onto solid substrate: description of hydrodynamic processes and heat transfers , J. High Temperature Material Processes, 1999, to be published.
[CAL 95]
CALTAGIRONE J.-P., KHADRA K. AND ANGOT P., Sur une methode de raffinement local multigrille pour la resolution des equations de Navier-Stokes , C. R. Acad. Sci. Serie II b, 320, 1995, p. 295-302.
[VIN3 99]
VINCENT S. AND CALTAGIRONE J.-P., One Cell Local Multigrid method for solving multi-phase flows , J. Comput. Phys., 1999, in corrections.
[RID 95]
RIDER W. J. AND KOTHE D. B., Stretching and tearing interface tracking problems , AIAA paper, 95, 1995, p. 1717.
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New Classes of Integration Formulas for CVFEM Discretization of Convection-Diffusion Problems E.P. Shurina and T.V. Voitovich Novosibirsk State Technical University, Novosibirsk, Russia ABSTRACT This paper is concerned with the development of FEM-like finite volume technique for discretization of convection-diffusionn problems over unstructured simplicial grids. Proposed finite volume technique is original in it's essence - technology of approximation of convection-diffusion fluxes and sources. Polynomial interpolation is used to represent unknown variables, ttransfer coefficients, thermophysical characteristics and source terms as linear combinations of local form functions at each element. Local form functions are expressed in terms of simplex barycentric coordinates. Three special classes of non-symmetric formulas for exact integration of monomials of barycentric coordinates over dual mesh lines and corresponding subregions are introduced. The first class contains formulas for exact integration of barycentric coordinate monomials along dual mesh lines lying on a simplex, and is used to express convection and diffusion fluxes in terms of discrete unknowns. The second class consists of formulas for exact integration of monomials of barycentric coordinates over simlplex subregions, corresponding to different finite volumes sharing the same element, and is used for discretization of source terms. The last class prescribes exact integration of the monomials along segments of the edges, approximating boundary curves, and is intended for realization of boundary conditions. Geometric interpretation of proposed integration formulas of the lowest order is presented. Described technology can be applied to a wide variety of PDE models, and with the use of volume barycentric coordinates allows uniform extension to three-dimensional case. Key Words: control volume finite element methods; unstructured grids; dual mesh; barycentric coordinates; upwind schemes.
1.
Introduction
Adequate numerical simulation of fluid and gas dynamics processes imposes strict requirements on the modern methods, namely: exact representation of domains with complex geometry and high-order accuracy of spatial and temporal discretization schemes. Two types of grids, non-orthogonal and unstructured allow
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exact treatment of complex domains. We use the latter to account for additional requirements, such as adaptivity and local grid refinement realization. As for the spatial discretization, the finite volume method has some advantages, in particular, conservative properties of corresponding schemes, simplicity and ability of more natural use of the upwind schemes. Formulation of a finite volume element scheme typically involves the following basic steps 1. A set of admissible discretization templates is chosen. 2. An arrangement of all unknowns of source nonlinear coupled system (cellvertex, cell-centered, side-centered arrangement) is specified. 3. Element based interpolation polynomials are specified for transfer coefficients, conservative variables, thermophysical variables and source terms. 4. Weighted residuals principle, with weighting functions being equal to one over each finite volume is applied. 5. For each element convection-diffusion fluxes through the dual mesh lines are expressed as linear combinations of local discrete unknowns, with coefficients of the combination being components of the local diffusion and convection matrixes. Similarly, integration of the source term interpolation polynomial over subregions of a finite element gives the local source vector. 6. Global matrixes and vectors are obtained via the element-by-element assembly. 7. Resulting sparse linear system is solved by a specified method. We are interested in the development of a finite-element like procedure for generation of discrete analogue coefficients, thus we will take a closer look at the fifth step. The main contribution of this paper lies in the development of a new uniform technique of convection-diffusion fluxes and source terms approximation, with exact integration of interpolation polynomials, which are represented with simplex barycentric coordinates. The technique is based on the three classes of special formulas introduced by the authors. These classes give exact values for integrals of barycentric coordinates monomials along dual mesh lines, simplex subregions and triangle edges segments. Alternative approach with exact integration of polynomials in finite volume discretization, in which all polynomials are expressed as a generalized Taylor series, was proposed by Yen Liu and Marcel Vinokur [LIU 97]. In FE discretization of Navier-Stokes equations only a few convergent pairs for pressure-velocity interpolation functions are known. In the case of triangular finite elements linear pressure - quadratic velocity is a usual choice. It has been shown that the Ladyzhenskaya-Babuska-Brezzi (LBB) restrictions still hold for hybrid FEM/FVM techniques [SHU 97]. To our best knowledge, appearance of LBB restrictions strongly depends on the choice of a pressure-velocity coupling technique. For example, SIMPLE-like algorithms [PRA 85] for linear interpolation functions of equal order do not lead to ill conditioned matrices for the pressure correction. Finite element community also reports several methods of circumventing the LBB restrictions [ZEN 95]. Thus, the triangular based linear form functions can be used to demonstrate the main features of the new approximation approach. However, the form functions of higher degrees are allowed, and this extension will be discussed later.
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2. Proposed finite volume technique Consider the conservation equation for a scalar quantity 0
where p is the fluid density, u is the velocity vector, F^ is the diffusion coefficient and SQ is the volumetric source of . Domain discretization. The computational domain is divided into triangular elements. Assume that computational points and finite element vertexes coincide ("the cell-vertex" arrangement), and both velocity components and scalar variables have the same finite volume ("the collocated" arrangement). Dual mesh is constructed on centroids of triangles and mid-points of the edges, such that each node / has the corresponding complete, or "incomplete" finite volume Q, , bounded by median segments (see Figure 1). Let Sf denote the boundary of £2,. Let us assume fixed global and local numeration on the set of triangulation nodes and on each finite element. The integral conservation form of equation (1) is
where J = Jc + Jd is the combined convection-diffusion flux of <j). The proposed CVFEM technology, similarly to the known ones, uses the following principles of finite element discretization: (i) specification of elementbased form functions for conserved quantities and sources (it should be noted, that the usage of interpolation functions by FV is more flexible, because different classes of profiles can be applied to different terms of equation (1)); (ii) element-by-element assembly of discrete analogue matrices and vectors. The essence of our contribution is the use of barycentric coordinates in local representations of scalar variables,
Figure 1. Fragment of primary (FE) and secondary (FV) grids corresponding to linear interpolation and notation used for a triangle en .
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transfer coefficients and sources and introduction of special integral formulas for barycentric coordinates monomials, much as in FEM. Consider a finite element en. Let 5"^ denote a segment of the dual mesh, lying on a median, which comes out of the triangle node nb (see Figure 1), and Jv nfr is a value of the combined flux through the median segment S^b in the direction of the outward normal to the boundary of the finite volume £lv , (see Figure 2). It will be sufficient to approximate three of the six introduced fluxes, because Diffusion
fluxes approximation.
Using Green's formula, the diffusion flux
through a median segment S^ can be represented as follows
It is well known that FEM naturally allows representation of significantly varying transfer coefficients by an interpolation function on an element, unlike most of the FVMs. Let us demonstrate proposed technology on a problem of approximation of the diffusion flux, using linear interpolation for F^ and 0 [3]:
Figure 2. Outward unit normals and corresponding notation for fluxes.
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where \j/m - iff m (x) are local linear basis functions (see Figure 3), Tm are the local values of F^ (m=l,2,3), and coefficients a^,b^,c^ can be expressed in terms of the local discrete unknowns 0i,02>03 coordinates of the triangular nodes:
va
'
an
mver
se matrix of the matrix of
[6]
Thus, approximation of the diffusion flux [3] becomes
It is clear from [7] that it is necessary to introduce special formulas for integration of the local basis functions along the median segments to complete discretization of the diffusion flux. Note, that in the case of triangular based linear interpolation, local basis functions coincide with simplex barycentric coordinates L | .,^ | .(x)=L | ., i = l,2,3. For every local basis function consider a pair of trapeziums and a triangle, which are constructed using finite element median segments and their images on the surface of the basis function (see Figure 3). Projections of the trapeziums and the triangle to coordinate planes gives
Figure 3. Some notation for the median projections and one of the local linear functions on the triangle en.
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Here lvx , /^ are the lengths of the projections of the median segment 5" onto the x and y axis, respectively, with the signs corresponding to the anticlockwise integration; v,nb& {1,2,3}. Taking into consideration formulas [8],[9] , we have for three determining diffusion fluxes of the element:
Finally, to determine coefficients in the first, second and third row of the local diffusion matrix, combinations 7^ - ^23' ^ 2 3 ~ ^ 3 l > ^31~^12 are considered in terms of conservation unknowns. Coefficients at 01? 02» 03 *n tne combinations give three elements in the current row of the local diffusion matrix. Approximation of convection fluxes is performed as
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Special Prakash-Patankar interpolation of u is used to avoid spurious pressure oscillations in the framework of the collocated approach. An upwind scheme is used to express (f)nb in terms of the local unknowns. The flux f „ pu-nds is \j> approximated using formulas [8,9]. Source terms approximation. Let Q.n, denote an intersection of the finite element en of interest and the finite volume of the node with local number nb
Figure 4. Notation for the finite element subregions Formulas for exact integration of barycentric coordinates over the subregions are introduced as follows
(One can easily obtain [14] representing each of the calculated volumes as a set of truncated right prisms). Formulas [14] are used to compute local RHS vector of en . Boundary conditions. For realization of boundary conditions (f.e. approximation of convection fluxes through outlet boundaries), one-dimensional analogues of [8,9] for integration of the local basis functions along the element edges segments are introduced. Future investigations. It is well known in the finite element community, that an advantage of the use of barycentric coordinates, is in the existence of integration formulas, simplifying computations of integrals along the edges of a triangular element and over an element itself, namely the following formulas by Eisenberg and Malvern[EIS73]:
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It is obvious, that in the proposed FV technology formulas [8,9,14] play the same role as [15,16] in FEM, and have the lowest order. Following classes of formulas for exact integration of barycentric coordinates monomials (integration over dual mesh lines, finite element subregions and boundary edges segments, respectively) are the subjects of our investigation:
Classes of the formulas [17] are the basis of the uniform discretization technique, which allows to form functions of higher order, since these functions can be represented through barycentric coordinates. Although the capabilities of [8,9,14], are such that these formulas of the first order can be used as a basis for CVFEM discretization, we consider the technique incomplete until the exact values for [17] are be found.
3.
Concluding remarks
A new approach for finite volume discretization is developed in analogy with the FE approach, namely we use barycentric coordinates in local representation of scalar variables, transfer coefficients and sources. Final discrete analogues of diffusion-convection fluxes and sources are obtained via special classes of integration formulas introduced by the authors.
4.
Bibliography
[LIU 97]
Liu Y., Vinokur M. Exact integrations of polynomials and symmetric Quadrature Formulas over Arbitrary Polyhedral grids, J.Comput.Phys., 140, 122, 1997.
[SHU 97] Shurina E.P., Voitovich T.V., Analysis of the finite element and finite volume methods based upon unstructured grids for solution of the Navier-Stokes equations, Computational Techlologies, 2,4, 1997 (in Russian). [PRA 94] Prakash, C, Patankar, S.V., A control volume-based finite-element method for solving the Navier-Stokes equations using equal-order velocity-pressure interpolation, Numer.Heat Transfer, 8, 259, 1985. [ZEN 95] Zienkievicz, O.C., Codina, R., A general algorithm for compressible and incompressible flow. Part I: The split, characteristic based scheme', Int.j.numer.meth. fluids, 20, 869, 1995. [EIS 73] Eisenberg M.A, Malvern L.E., On finite element integration in natural coordinates, Int.j.numer.methods eng., 1, 574, 1973.
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Analysis of Finite Volume Schemes for Two-Phase Flow in Porous Media on Unstructured Grids
M. Afif (1) and B. Amaziane(2) (1) Universite Cadi Ayyad Faculte des Sciences Semlalia B.P. S.15, 40000 Marrakech Maroc. E-Mail: [email protected] (2) Universite de Pau et des Pays de I'Adour Departement de Mathematiques Recherche Av. de I'Universite, 64000 Pau France. E-Mail: [email protected]
ABSTRACT This paper is devoted to the analysis of finite volume schemes on unstructured grids for a nonlinear, degenerate, convection-diffusion equation arising in flow in porous media. A semi-implicit scheme is considered. We prove that this scheme is L°°, weak BV stable under a CFL condition and satisfy a discrete maximum principle. We then derive a convergence result. Results of numerical experiments using the present approach in the 2-D case are reported. Key Words: finite volume method, degenerate parabolic equation, nonlinear diffusion-convection, porous media, unstructured grids.
1. Introduction Flow simulation in petroleum and groundwater reservoirs has been extensively studied using finite element methods in past years (see, e.g., [AWZ 96, CHJ 86, CHE 97] and the bibliographies therein). Also, a discretization using both finite element and finite volume methods for two-phase flow in porous media is presented in [CJR 95]. However, it appears that there are few results on convergence theory for the degenerate problem [AWZ 96, CHE 97]. More
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recently, finite volume methods were developed and analyzed for immiscible two-phase flow in porous media in the case where the diffusion term is neglected, (see [EGH 98, VEV 98] and the references therein). This approach leads to robust schemes applicable for unstructured grids and the approximate solution has various interesting properties which correspond to the properties of the physical solution . These methods have been useful for advective flow problems because they combine element by element conservation of mass with numerical stability and minimal numerical diffusion. In this paper we will study a numerical approximation of two-phase flow in porous media. We focus on two-phase immiscible flow, which corresponds physically to the water flooding of a petroleum reservoir. We consider twophase water and oil flow in a porous media, using the global pressure, the total velocity and the water saturation as the primary variables. This formulation leads to a coupled system of partial differential equations which includes a nonlinear degenerate parabolic saturation equation and an elliptic pressurevelocity equation [CHJ 86]. The saturation equation is convection dominated and thus special care should be taken in discretization. The diffusion term there is small but important and can not be neglected. Also irregular geological features, which suggest the use of irregular grids. We will consider a system of equations describing the flow of two immiscible incompressible fluid phases through a porous medium in which for sake of simplicity, we will neglect the effect of gravity. This system is then the following: Saturation Equation:
Pressure Equation:
where u(x,t) is the water saturation, P is the global pressure, K(x) is the absolute permeability tensor of the reservoir Q C R2 whose boundary F splits up into three parts such that F = FI U ^ U Fa and F,- D Fj = 0 for i ^ j; FI is the part of the boundary where the water is injected, F2 is the impervious part of the boundary and T3 is the producing part of the boundary. Also Qr denotes £l x ]0, r[.
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the saturation equation that satisfy the discrete maximum principle and derive convergence results. More precisely, we will consider a numerical approximation of this system where the saturation and pressure equation are decoupled. We will concentrate on the study of the convergence of the FVS for the saturation equation taking into account the diffusion term and the anisotropic heterogeneity of the reservoir. A mixed-hybrid finite elements method is used to obtain an accurate approximation of the velocity [BRF 91]. In [AFA 97], the one-dimensional problem was analyzed. In this paper, we extend these ideas to multiple dimensions. We will restrict attention to R2, however, the methodology and the analysis can be extended to problems in R3. This paper is organized as follows. In the next section, we present the finite volume discretization of the problem (P1). An explicit approximation of the convection term combined with an implicit approximation of the diffusion term is considered. The solution is approximated by a piecewise constant. In section 3, we present the L°°, weak BV stability under a CFL condition and a convergence result. Numerical simulations for a 2-D example are presented in section 4. 2. Finite volume discretization
Before describing the finite volume discretization of the model problem (P1) we give some notations. As usual, let ( t n ) n = Q Nr be a partition of J with a time step At = i n + i — tn and let A/j = (Ti)i=o Ne be an admissible triangulation of fi, S/j = (Mi)i=00NS the Donald dual mesh, whereere 2 h := min|M;| verify sup|M,-| < S.h2, and XT the barycenter of T E Ah, is i
i
such that XT -
f| MnT^0
dM £ T. We denote by XM :=
U
dT £ M.
TnM^0
l := dMi U dMj Pi T the line segment between the points XT and Xij the midpoint of (XM, ,XMj) and let £h — {I £ dM\F , forM £ Eh}- For an initial condition u° 6 L°° (Q) n BV (ft), we set u°M = rj^ fM u° (x) dx. For simplicity we assume that the functions 3> and K are piecewise constants, we set $M — *^|M and KT = K\T- Let u^ be an approximation of u ( x M , t n ) which will be defined precisely in the sequel Integrating (1) over the set M x [t n ,tn+1] where M £ Eh,, we obtain the following scheme
where nM,I is the outward normal to / £ dM. Using an explicit approximation of the convection term and an implicit approximation of the diffusion term, we
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get
which could be written in the following form
where
with MI E Eh is such that / € MI D M. The advection term is approximated by an upwind Godunov scheme and the diffusion term is approximated in the following way: for T PI M = 000
where L £ dT such that LDM = 0, and using a piecewise linear approximation for Va, we get
where XM 6 P1 such that XM,( x Mj) = &ijFinally, since divq = 0 we have the following semi-implicit scheme
where DM,i = - \T| VXM,,T • AWxM.T for all / G dM\T.
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Remark 1 We can use an explicit fresp. implicit] approximation for both the diffusion and the convection terms to obtain an explicit [resp. implicit] finite volume scheme which lead generally to large CPU time computing. For more details on the comparaison of these schemes see [AFA 97].
3. L°° stability weak BV estimates and convergence results In this section we shall analyze the F.V. scheme obtained in section 2. Let us state the following assumptions: is a bounded open polygonal subset of R2. such that - (A3) K is a bounded, uniformly positive definite symmetric tensor on Q
- (A4)
such that
- (A5)
such that
- (A6) b,d
and
such that 6 is a monotone increasing function, and
It should be noted that a full tensor for the absolute permeability K is considered and the approximation obtained in (4) is such that 0 < D_ < DM,i < D+ < oo which is an important property for the analysis of this FVS. In fact, we can write the coefficient DM,I in the following forme:
where L and L' G &T such that L D M = 0 and L' U MI - 0, hence
Furthermore, we have DM,I = -^^X'XM • adj [KT]X'^MI where x' — L fl L' and adj is the adjugate matrix. Now adj [/\T] is also a symmetric definite positive matrix, then it has a unique factorization, adj [KT] — \Ur] [UT], in which UT is upper triangular with a positive diagonal. We have
then for K(x) = k(x)Ko in £7, where k(x] > 0 and Ko is a constant symmetric rrt positive definite matrix such that adj [Ko] = [Uo] [Uo], we can triangulate
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QKo := [U0] (£2) a transformation of the domain fi by the matrix Uo, with an admissible triangulation such that
where TJ is a small parameter independent of h, then we get also 0 < D_ < DM,l • The extension of the present technique to the case where
such that
K(x)|n, p = k p ( x ) K p , with kp (x) > 0 and Kp is a constant symmetric definite positive matrix, for all p = 1,.., Np, is straightforward. Let Cq defined by
where NM0 = card {l E 9M}, such that sup N0M is assumed to be finite, we M introduce the CFL condition defined by:
The following results hold: Proposition 1 Under the assumptions (A1)-(A7) and the CFL condition (5) the scheme (4) is L°° stable. Moreover, the approximate solution (unM} satisfies
Proposition 2 Under the assumptions (A1)-(A7) and the CFL condition (5) with [CFL < I — e], we have the following estimates
where s is a small parameter and C is a constant independents of h and At. Let us introduce a weak solution u of the problem (P1):
The data are assumed to be smooth enough to guarantee the existence and uniqueness of the weak solution. We have the following result: Theorem 1 Under the assumptions (A1)-(A7) the approximate solution Uh, given by the scheme (4) and the CFL condition (5) with [CFL < 1 — e\, converges to u in L2(QT) as h and At goes to zero.
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Figure 1. Reservoir &
Figure 2.
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Triangulation of QQ = UQ (&}
5. Numerical simulations
In this section, we present some numerical results in 2-D. An IMPES simulator which applies a mixed-hybrid finite element method for computing pressure and fluid velocity approximations and the finite volume scheme described here for the saturation was developed. Numerical results prove the effectiveness and the robustness of this methodology. We have run various simulations, herein we are presenting only one of such a simulation to illustrate our results. A simulation was performed on a heterogeneous reservoir shown in Figure 1 for a two phase flow problem. An injection well is placed at the lower left hand corner of the reservoir and a production well is placed at the top right hand corner of the reservoir. The highest value of permeability (K = 10) is in the darkest shaded blocks. The lowest permeability (K = 0.1) is in the lightest shaded blocks. The intermediate value of permeability (K — 1) is the intermediate shade in the picture.
Figure 3. Saturation contours
Figure 4. Total velocity q
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Finite volumes for complex applications
6. Bibliography [AFA 97]
AFIF M. AND AMAZIANE B., On convergence of finite volume schemes for one-dimensional two-phase flow in porous media, Preprint, 1997, to appear.
[AWZ 96]
ARBOGAST T., WHEELER M.F. AND ZHANG N.Y., A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Num. Anal., 33, N° 4, 1996, p. 1669-1687.
[BRF 91]
BREZZI F. AND FORTIN M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
[CHJ 86]
CHAVENT G. AND JAFFRE J., Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986.
[CJR 95]
CHAVENT G., JAFFRE J. AND ROBERTS J.E. , Mixed-hybrid finite elements and cell-centred finite volumes for two-phase flow in porous media, in: A. Bourgeat, et al., eds., Proceedings Mathematical Modeling of Flow Through Porous Media, World Scientific, London, (1995), p. 100-114.
[CHE 97]
CHEN Z. AND EWING R.E., Fully-discrete finite element analysis of multiphase flow in ground water hydrology, SIAM J. Num. Anal, 34, N° 6, 1997, p. 2228-2253.
[EGH 98]
EYMARD R., GALLOUET T., GHILANI M. AND HEREIN R., Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes, IMA J. Num. Anal., 18, 1998, p. 563-594.
[VEV 98]
VERDIERE S. AND VIGNAL M.H., Numerical and theoretical study of a dual mesh method using finite volume schemes for two phase flow problems in porous media, Numer. Math., 80, N° 4, 1998, p. 601-639.
A preconditioned finite volume scheme for the simulation of equilibrium two-phase flows
Sebastien Clerc Commissariat a I'Energie Atomique, CEA-Saclay, 91191 Gif-sur-Yvette FRANCE
ABSTRACT In this paper, we report on a preconditioned finite volume technique to simulate two-phase flows in complex geometries. Key Words: two-phase flows, low Mach number flows, preconditioning.
1. Introduction
The numerical simulation of two-phase flows is a crucial problem for the design and safety analysis of heat exchangers, especially in the nuclear industry. Generally, the fluid is in the liquid phase at the inlet of the exchanger. Because of the heat release or a loss of pressure, vapor is created. The description of the interface between the phases would require a very fine space scale with respect to the characteristic length of the geometry. For this reason, one must consider a macroscopic model. This model should reproduce some of the characteristic features of two-phase flows. The most basic of these features is probably the compressibility effect introduced by vapor bubbles. Indeed, although the density of liquid water can usually be considered as a constant, the apparition of bubbles creates large variations of the mean density of the flow. Moreover, the dependence of this mean density with respect to the pressure can not be neglected. In other words, if we consider the bubbly flow as a homogenized mixture, the resulting averaged speed of sound is quite low in usual conditions. In order to study this feature, we consider the very simple homogenizedequilibrium two-phase flow model. Indeed, using more sophisticated models would introduce other difficulties which are not in the scope of this study.
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Finite volumes for complex applications
2. The equilibrium two-phase flow model
The homogeneous equilibrium model of liquid/vapor flows is characterized by a kinematic and thermodynamic equilibrium assumption. The evolution of the mixture can be described by the Euler equations for a single fluid:
Here E = e + |u| 2 /2 denotes the total energy, and H = h + |u| 2 /2 the total enthalpy of the mixture. To close the system, an equation of state (EOS) links the pressure p to the conservative thermodynamic variables p and pe. The pressure law must be such that the partial derivatives x and K with respect to p and pe satisfy The sound speed c of the fluid is the square root of this quantity. Generally, a drift flux is added to the set of equations (l)-(3) to incorporate the effects of the slip velocity between the phases. This will however not be the case in the present study. 2.1 Equation of state The equation of state consists of three zones: a liquid, a two-phase mixture, and a vapor zone. The liquid and the vapor zones are described by usual single fluid equations of state. In the two-phase zone, each fluid is supposed to be at saturation, so that the density pt,pv and the enthalpy hi,hv of each phase are functions of the pressure only. The density p and the enthalpy h of the mixture satisfy :
The transitions between the mixture zone and the two others occur along the saturation curves. The equation of state is continuous across this curve but not continuously differentiable. This fact is clearly visible in Fig. 1.
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Figure 1: Specific volume l/p versus enthalpy h for the water equilibrium EOS, at pressure p = IMPa. The liquid zone corresponds to lower values of h, and the vapor zone to the higher values of h. In the two-phase zone in-between, 1/p varies linearly with h. Note that the derivatives of the equation of state are discontinuous across the transitions. 3. The preconditioned finite volume scheme 3.1. Some notations
We rewrite the Euler equations (l)-(3) with the compact form:
We consider a triangulation of the computational domain by polygonal cells. For a cell A', we denote by |K | its volume and N ( K ) the set of neighboring cells. If J E N ( K ) , the surface of the common interface is (TKJ and *IKJ denotes the unit normal, oriented from K to J. Finally, we denote by 6t the step of the time discretization. The finite volume method for the solution of the system of conservation laws (4) takes the form:
Here, U% denotes the average value of the solution in the cell K, at time nSt. To completely determine the numerical method, we have to specify the expression of the numerical flux <&K J in terms of the cell average values. We will consider
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Finite volumes for complex applications
numerical flux functions of the form:
The viscosity matrix Q depends on the left and right states UL and UR. The numerical flux we have used is a preconditioned version of Roe's numerical flux. We will first begin by describing the extension of Roe's flux to our system, and then introduce the preconditioning. 3.2. Roe's flux Roe's scheme is obtained by chosing for Q = |A|, where A is the Roe matrix, and |A| = ^ |A,-|r,-(g)l,-, if A = ^ A,Tj<S>l,- is the eigen-decomposition of A. We will denote by ([•] the difference between the right and left values of any quantity, i.e. (•)# — (-}L- The Roe matrix has to satisfy the following properties: • An has a complete set of real eigenvectors • A n (t/L, UR) -)• DF(U) • n when UL and UR tend to U. • An(ULtUR)lU] = [ V ( U ) - u ] . As is well known (see e.g. [ViMo, To 91]), a Roe matrix for the Euler equation with an arbitray EOS can be found if we can provide a linearization of the pressure jump such that
Geometrically, the solutions to this problem are located on the intersection of a line and a half plane in the (x, K) plane. We will denote by [x] the vector ([/?], fl/>e]) and by Dp = (x, K), such that (7) becomes [p] = [x] • Dp. Following the lines of [ViMo], we start with a point Dp which is some average value of DPL and D p R , but does not satisfy (7). We project it on the line, with respect to a scalar product defined by a given symmetric matrix M:
We can then check a posteriori if condition (8) is satisfied. If not, we must consider another initial guess Dp. It is clear that if p is continuously differentiate, £ stays bounded when [x] tends to zero. Moreover, if we choose for M_an approximate value of the Hessian of p, then £ will be close to one. Finally, Dp will tend to Dp as expected. For
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our simulations at p = 5 MPa, we found that a diagonal matrix diag(l, 10~ 12 ) gave satisfactory results. However, if we are close to a point where p is not continuously differentiable, then £ will certainly blow up when |x| goes to zero. As a result, the value of Dp will not tend continuously to Dp. This fact lies at the root of some numerical difficulties in the computation of phase transition inside the flow. It becomes particularly important when the discrepancy between the derivatives is large, i.e. for lower values of the pressure. Obviously, more research seems in order on this point.
3.3. The preconditioned flux For low Mach number applications, it is preferable to use a preconditioned version of Roe's scheme. The viscosity matrix (see eq. (6)) is now Q = P-1 |PA|, where P is a properly chosen non-singular matrix. Note that since we have only modified the viscosity matrix, the consistency of the numerical scheme is not affected. Therefore, the scheme is time-accurate and suitable for transient problems. This formulation was first introduced by [GuiVio] and further studied by [Cl 98], see also [PCV+]. In this work, we have used Turkel's diagonal preconditioner [Tu 87]. This matrix depends on a single parameter B which was set to the local Mach number.
4. Implicit time stepping
4.1. Choice of the variables for the linearization In low Mach number applications, an implicit time stepping cannot be avoided. First, the CFL constraint on the time step for explicit schemes becomes too restrictive when the sound velocity gets large. Moreover, for the time-accurate preconditioned scheme, the stability condition is even more restrictive than the CFL condition. One can consider the usual linearized backward Euler scheme:
with It is also possible to linearize the problem with respect to an other set of variables: setting U — U(V), one can consider
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Finite volumes for complex applications
with Vn+1 = Vn + 6V. At steady state (6V = 0), this formulation is still conservative. The conservation is not ensured during the transient but can be restored if necessary by performing a few Newton steps within a time-step. Note that this is also true of the conservative variables formulation (9). More specifically, we advocate the use of the following set of variables: ( p , p u , p ) . Including the pressure in the set of variables has two advantages in our case. First, it makes the computation of the thermodynamic variables simpler and faster. If the density and the pressure are known, we can readily know in which zone of the EOS the state is located. On the contrary, if the energy rather than the pressure is known, we must first guess in which zone it is, and modify this guess if necessary. More importantly, it can dramatically improve the convergence of steady state computations when contact discontinuities are present (see [Cl 99]). Indeed, the pressure time-increment is usually very small in low Mach number flows, although the density and the energy time-increments can both be important. With a non-linear EOS such as the equilibrium EOS, the conservative variables formulation (9), the density and energy increments will automatically induce spurious large pressure increments. This phenomenon is responsible for the apparition of spurious acoustic waves of large amplitude in the solution, which prevents the use of large time steps in the implicit scheme. With formulation (10), larger time steps can be used. Although in some cases the time step can be set close to infinity, it is generally advisable to stay close to 6x/u, where 6x is the typical mesh size and u is the reference material velocity. In low Mach number flows, this time step is of course much larger than the maximum explicit time step. 4.2. Approximate Jacobian for the preconditioned flux Equation (9) (and (10)) require the computation of the partial derivatives dO/dUL and dQ/dUR of the numerical flux. Due to the algebraic complexity of the numerical flux, an exact computation of these matrices is not easy, especially for a general EOS. We have thus employed the usual approximation for Roe's flux:
To obtain the derivatives with respect to V, we left-multiply these matrices by the Jacobian of the change of variables dUfdV. Practically, it amounts to expressing the left eigenvectors of A (for Roe's scheme) or PA (for the preconditioned scheme) in terms of the pressure rather than the energy.
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5. A two-phase bump channel flow.
We briefly present a two-dimensional test-case for the equilibrium two-phase flow model. We consider a channel with a 20% sinusoidal bump (Fig. 2). A
Figure 2: Channel with bump: 20 x 80 structured mesh. pressure of 5 Mpa is imposed at outflow. A constant velocity of 10 m • s-1 is imposed at the inlet, and the enthalpy is set to 1154.2 KJ. This corresponds to a liquid state close to the saturation. As the pressure drops with the restriction of the section, a small concentration of vapor appears. The Mach number jumps from 10- 2 to 0.4, so that the compressibility becomes important. We initialize the computation with a purely liquid steady-state computed with a higher value for the pressure. The pressure profile is displayed on Fig. 3. As expected, no discontinuity can be detected across the phase transition lines. On the contrary, the influence of the vapor is clearly visible on the Mach number profile (Fig. 3, right), which shows sharp discontinuities. For both profiles, the symmetry is quite satisfactory.
Figure 3: Channel with bump: Left: pressure, 20 isolines. Right: logarithm of the Mach number, 80 isolines. Although the Mach number exhibits a strong discontinuity, the pressure field is continuous.
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Finite volumes for complex applications
6. Bibliography
[Cl 99]
CLERC S., Accurate computation of contact discontinuities for flows with general equations of state, Computer Methods in Appl. Mech. Engrg., to appear.
[Cl 98]
CLERC S., On the preconditioning of finite volume schemes, 7th Int. Conf. on Hyperbolic Problems, Zurich, February 1998, Birkhauser.
[GuiVio]
GUILLARD H., C. VIOZAT, On the behavior of Upwind Schemes in the Low Mach number limit, Computers & Fluids, 28, 1999.
[PCV+]
PAILLERE H., CLERC S., VIOZAT C., TOUMI I., MAGNAUD J.-P., Numerical methods for low Mach number thermalhydraulic flows, Proceedings of the 4th ECCOMAS Conf. on Comput. Fluid Dynamics, Athens, 1998. John Wiley & sons.
[To 91]
TOUMI I., A weak formulation of Roe's approximate Riemann solver, J.C.P., 102, 1991, pp. 360-373.
[Tu 87]
TURKEL E., Preconditioned methods for solving the incompressible and low speed compressible equations, J.C.P., 72, 1987, pp. 277-298.
[ViMo]
VINOKUR M., MONTAGNE J.L., Generalized flux-vector splitting and Roe-average for an equilibrium real gas, J.C.P., 89, 1990, pp. 276-300.
Transient flows in natural valleys computed on topography-adapted mesh
S. Soares Frazao, J. Lau Man Wai, Y. Zech Universite catholique de Louvain Place du Levant 1 B - 1348 Louvain-la-Neuve
ABSTRACT The paper deals with the finite-volume computation of severe transient flows in natural valleys. The major problem concerning those flows is to have an appropriate representation and treatment of the topographical source term. A lateralization of the intercell flux is presented here to account for the bottom slope. Computation efficiency is gained by using a topography adapted mesh. Finally, an application to a Belgian valley is shown. Key Words: Finite-Volume, topography, source terms, mesh
1. Introduction Severe transient flows in natural valleys occur during extreme flood events, or as a result of a dam-break, which can be considered as the worst case. Several Finite-Volume numerical schemes have been developed for the computation of such flows, mainly focusing on an accurate evaluation of the numerical flux between cells. However, in natural topographies, important source terms appear due to the shape of the valley itself, they can also have relevant effects on the wave propagation. Therefore, an important step for a reliable computation is thus an accurate representation of the valley complexity, requiring a suitable mesh generation, in order to achieve a proper calculation of the topographical source term.
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Finite volumes for complex applications
2. Numerical scheme
A Finite-Volume numerical scheme is used to solve the Saint-Venant shallowwater equations. Those are written, in the 2D case,
where subscript t and x denote time and space derivatives respectively. The vectors of dependent variables £7, fluxes F(U) and G(U), and source terms S(U) are defined as
with h the water depth and it, v the depth-averaged velocity components. So and Sf are the bottom and friction slope, respectively written for the x- and y-direction. Friction slope Sf can be evaluated from the Chezy, Manning or another empirical formula. Integrating (1) on a discrete cell yields the following finite-volume scheme
where A is the cell area, Lj is the length of the cell interface j and n the number of cell interfaces. The vector U is the vector of rotated hydraulic variables obtained by applying the rotation matrix T to the original vector U and is aligned with the new co-ordinates axes (x , y ) where x is the direction normal to the considered cell boundary. The numerical fluxes between two cells located at the left and right side of the interface (cells / and r) are then calculated by the Boltzmann model [SOA 99]
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Figure 1: Lateralization of the topographical source term where Fr is the Froude number. Fr represents the local Froude number, calculated with the velocity normal to the interface. 3. Lateralized topographical source terms
In each discrete cell, the bottom level zb is assumed constant. To compute the topographical source term S0X = — ^, the concept of flux lateralization [CAP 96] is used, and adapted to 2D problems. This means that the flux coming out of the cell located at the left side of the interface differs from the flux entering the cell located at the right side by a quantity gdz(hieft — ^-) accounting for the change in the bottom level (see figure 1, hatched area). The hydrostatic pressure term g^- in the flux is calculated with h being (hieft — dzb} or (hright) for the left and right cell respectively. This ensures that for water at rest, the pressure terms and the topographical source term, which are the only non-zero terms, cancel each other out. According to Nujic's compatibility condition [NUJ 95], the water will thus stay at rest. The lateralization is a very robust method, as no special reduction of the time step (or of the CFL number) is needed. However, it must be stated that it is probably not the most accurate method, especially for 1D problems. Indeed, in some cases, influence on the wave front velocity was observed, leading to a too slow flow propagation. 4. Topography-adapted mesh generation
The description of the valley topography coming from a DTM (Digital Ter-
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Figure 2: Definition of the segment density rain Modeling) usually consists of a square grid of points giving the bottom elevation. For dense grids, it is likely that the computation time becomes too important when considering all the available points. On the other hand, building a uniform and coarser mesh upon the given square grid can lead to accuracy losses and interpolation errors. In fact, rather flat areas could be represented with a lower mesh density than regions with abrupt changes in the topography and local steep slopes. A triangular mesh based on a given DTM can be constructed as follows. A first triangulation is generated only with the points defining the boundary of the computational domain. Additional intermediate points are then added one by one on existing segments, until the requested refinement has been reached. The mesh segments to be divided are selected according to their density. A segment between two points A and B (see figure 2) is said to have a high density when the error made by replacing the exact topography between A and B by a straight line is important. The error is quantified in the following way :
After each addition of a new point, the mesh is checked locally to avoid triangles with small internal angles. This leads to a local Delaunay mesh, with well conditioned triangles, and a finer discretization where needed, according to the given topography. 5. Application to the valley of Robertville The valley of Robertville is situated in the Eastern part of Belgium. A concrete gravity dam is located at the upstream end of the computed reach. The valley is steep and narrow, and widens at the downstream end, near the city of Malmedy (see figure 3). The aim of the study that was carried out was to determine the arrival time
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Figure 3: valley of Robertville - distances in [m] of the water at the city of Malmedy in case of a dam break. A DTM was available for the valley, the grid size being of 40m. 5.1 Estimation of the friction
coefficient
A Manning friction coefficient of 0,02 sm- 1 / 3 was estimated for the valley. It is clear that a unique friction coefficient for the complete valley is not the most accurate choice, but this was due to the lack of information. A sensitivity analysis was thus carried out and showed the importance of this parameter on the wave propagation velocity. 5.2 Computed results The failure of the dam is considered as instantaneous. Several computations were run, and the criterion selected to compare the results is the arrival time at the city of Malmedy. A first run with a ID model gave a very short time (see Table 1). Computation were then run with different triangular mesh refinement levels identified by the maximum cell interface length : 100m, 70m 60m. The square grid given by the DTM was also used to compare with the other meshes. Figure 4 shows the inundated area once the water has reached the city of Malmedy. The city is located in the bottom left corner of the figure and the upstream reservoir is located at the right side of the dam. The sharp bends and the very narrow shape of the valley clearly appear. It can be seen in Table 1 that for triangular meshes, the travel time decreases
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Figure 4: Inundated area - distances in [m] with the mesh refinement. However, it is not expected to reach the short travel time of the ID model, even with further refinement. Indeed, the ID model considers the valley as being straight, i.e. without any bends, which is absolutely not the case for Robertville. The influence of bends on the flow was shown in [SOA 99]: important head losses occur, that slow down the wave front considerably. Sharp bends can even lead to partial reflection of the flow, with a bore travelling back in the upstream direction. The square grid appears to lead to the longest travel time. A possible explanation is the important error made on the flow path when the valley is not aligned with the mesh (see figure 5). A similar feature can be observed on mesh
time {minutes}
1D
6 17 21 31 50
2D triangulaaar (L = 60 m)
2D triangular (L = 70 m)
2D triangular (L = 100 m) 2D squaaree
Table 1: Arrival times at Malmedy
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Figure 5: Influence of the square grid on the flow path triangular meshes, as the cell limits do not always match the exact wetted area delineation. However, with a triangular mesh, the error is less, as no preferential flow path is induced. Moreover, the fact that the mesh is generated according to the topography leads to a denser mesh at the steep river banks, allowing a more accurate representation of the wetted area delineation. 6. Conclusions
A lateralized treatment of the topographical source term of the 2D shallowwater equations was presented. Computations of a dam-break flow in a steep natural valley were shown on different kinds of meshes (triangular and square). The importance of an accurate representation of the topography of natural valleys clearly appears : the mesh used has a significant influence on the wave front velocity. Indeed, important differences in terms of wave propagation time were observed on the different runs. References
[SOA 99]
SOARES FRAZAO S., SILLEN X., ZECH Y., Dam-Break Flow Through Sharp Bends. Physical Model and 2D Boltzmann Model Validation , to appear in CAD AM, Proceedings of the first expert meeting, Wallingford UK
[CAP 96]
CAPART H., YOUNG D.-L., HUANG S.-Y., PAN J.-M., A lateralized flux-predictor scheme for a computation of openchannel flow in arbitrary topography , Proceedings of the 20th
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National Conference on Theoretical and Applied Mechanics, National Taiwan University, Taipei, China (1996) [NUJ 95]
NUJIC M., Efficient Implementation of Non-oscillatory Schemes for the Computation of Free-Surface Flows , Journal of Hydraulic Research , 33 (1), 1995
A mixed Finite Volume/Finite Element method applied to combustion in multiphase medium
Nathalie Glinsky-Olivier +, Eric Schall + + CERMICS-INRIA, Sophia Antipolis, France, + INRIA, Sophia Antipolis, France
ABSTRACT The present study is concerned with a mixed finite volume/finite element method applicable to triangular unstructured meshes for the simulation of wildfires propagating through a fuel bed using the multiphase approach. The forest fire is represented by a gas phase propagating through solid phases representing the heterogeneous combustible medium. Attention is focused on the development of a solution method of these equations. An explicit second-order in time and third-order in space accurate scheme has been constructed based on a special approximate Riemann solver applying the Roe's formulation. The radiative energy flux is calculated by the Discrete Ordinates Method. This model has been applied for simulating a wildfire through a litter of dead pine needles. Key Words: finite volume, finite element, multiphase approach, upwind scheme, combustion.
1. Introduction The present study is concerned with simulation of wildfires propagating through a fuel bed using a multiphase mathematical model. For this reason, attention was focused on an adaption of the Roe's scheme to our new set of equations. The source term includes many complex physical phenomena (heating, drying, pyrolysis of the fuel material and char combustion) and operates in each conservation equation.
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2. The physical model For this study, we have applied the multiphase formulation proposed by M. Larini et al. [LAR 99]. A brief description is given below. The fuel bed is supposed to be an heterogeneous medium made of solid particles of different types. A small control volume V contains N solid phases and a gaseous phase. Each solid phase is composed of particles of same thermochemical and geometrical properties (i.e. shape, size, arrangment) providing the same behavior under fire. The gas phase is a mixture of five species : CO, CO2, H2O, O2 and N2 numbered from 1 to 5 in this order. The packing ratio of the phase k is cx.k — Vk/V where Vk is the volume occupied by the phase k in the total volume V. In the same way, the fractional porosity is ag = Vg/V. We then have the relationship ag + X}fc=i ak — 1For the gaseous phase, we consider the conservation equations for the variable Wg where p — pg ag and pg is the density of the gas, (u, v] is the velocity of the gas, e is the specific internal energy and Yi are the mass fractions of the species with ^j=1 YI — 1. This relationship allows us to solve only 4 conservation equations for the species, the fifth mass fraction being deduced from the others. The set of conservation equations has been provided by M. Larini and B. Porterie [FOR 98] following the ideas in [GRI 85]. It can be written in conservative form :
where the notation Wt means dW/dt. The convective fluxes F and G write :
The pressure agp is calculated by using the perfect gas law for a gas mixture and the definition of the specific internal energy
where M; is the molar mass of the species i and Tg is the temperature of the gas. By combining these relationships, the pressure can be expressed as a function
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of 7, the ratio of the specific heats for the gas mixture
The diffusive fluxes R and S write:
where T> is the diffusion coefficient of the species. The stress tensors have the classical expressions
is the source term :
where riiH2o represents vaporization, mpr and msurf are the the solid mass loss due to pyrolysis and char combustion respectively. dragx and dragy are the two components of the drag force and g is the norm of the gravity. Qrad and Qcond represent the heat transfer between gas and solid phases due to radiation and conduction respectively. Ahi0 are the formations enthalpies of the species. More details can be found in [FOR 98]. The solid phase is described by the variable Wk [FOR 98] Wk ==• (ctkpk,Tk) where pk and Tk are respectively the density and the temperature of the solid phase A;. The mass balance and the heat conduction equations for the solid phase k write :
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where Cpk is the specific heat at constant pressure and Tfc is the temperature of the solid phase k. The solid phases are supposed to be at rest, then the momentum equations reduce to u^ = Vk — 0. These equations contain no flux, only a source term. The equations for the solid phase are solved separately from the gaseous phase equations both system being coupled by the source terms S19 and fi^ which depend on Wg and W&For the closure of the model, we distinguish two types of boundary conditions : the open sides (right, left and top of the domain) when a special upwinding is applied, and the lower boundary of the domain which corresponds to the ground where we apply no slip conditions for the velocity u = v = 0 and adiabatic temperature dT/dn = 0. The system formed with the convective fluxes is hyperbolic i.e. for any vector (771,772), the matrix M = r)idF/dWg -}-^dG / dWg is diagonalizable. Its eigenvalues are real and are
Then, we can write M = T-l(Wa,fl}\(Wg,iif}T(Wg,jf) where A = diag(\i] is the diagonal matrix composed of the eigenvalues A; of M and T is the invertible transformation matrix. The absolute value of M. writes \M\ — T~1(Wg,ff) \A.(Wg,ff)\T(Wg,f)). We can notice that the two last eigenvalues Ay and AS have different expressions from the classical reactive Navier-Stokes system. Their values are close to the usual eigenvalues u TJI + vr\2 ± \/7 p/p.
3. Spatial approximation We give here a short description of the numerical model, details can be found in [SCH 99]. Starting with a (possibly unstructured) Finite Element triangulation of the calculation domain D, dual Finite Volume cells are constructed joining successively the gravity centers of the triangles having a node i as a vertex and the middle of the sides adjacent to this node. We write the variational formulation of the gaseous system. The mixed FV/FE formulation traduces by the choice of the test functions \l>;. For the temporal, convective and source terms, the test function \I>; is the characteristic function of the cell Ci and for the diffusive part, \&; is the PI Galerkin basis function at the node i whose support is Si = UT,iETT . After integration of the different terms on their respective support, application of the Green formula and mass-lumping
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for the temporal and source terms, we obtain
rri
where Ai is the area of the cell Cj, (n^n^} is the outward normal to the cell Ci and (nx,ny) is the outward normal to the boundary of the calculation domain D. A first-order upwind approximation of the convective fluxes write
where % is the integral normal to the interface between CT and Cj and is the numerical flux through this interface. This numerical flux is calculated by applying a Roe scheme [ROE 81] adapted to our special convective fluxes F and G. For the calculation of the fluxes of the mass fractions equations, we use the multi-componenents flux proposed by B. Larrouturou [BLA 89], which allows to preserve positivity for the mass fractions. IfOPijis the numerical flux for the density variable, the flux of the species Y is :
High-order spatial accuracy is obtained via the M.U.S.C.L. method [VLE 72], [LFE 85] and a /3-scheme formulation which consists in writing the numerical flux $ (Wij.Wj^ffij) with for instance W%j = Wi + 1/2[(1 - 2/3)(W3 - WJ + 2(3VWi.ij where (3 is an upwinding parameter. Choosing (3=1/3 provides a third-order accurate scheme in space. A PI Finite Element interpolation for the approximation of the diffusive fluxes leads to
where AT is the area of the triangle and R\T and S\T are constant on T. More details can be found in [LFE 89]. The multiphase radiative transfer equation is solved using the Discrete Ordinates Method [SAK 96]. Since the soots are not included in this model, the contribution of the radiation is negligible compared to the other physical phenomena. A detailed description of this implementation can be found in [SCH 99].
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3. Numerical results This numerical model has been applied for describing the behaviour of a spreading wildfire through a litter of dead pine needles. The domain is 1m height and 2m long and the fuel depth is 0.04m. The mesh is composed of 5151 nodes. Test conditions are the same as described in [POR 98]. The fire is ignited at the middle of the litter and the ignition temperature of the solid is equal to TOOK since the gas is at ambient temperature. We present the isovalues of the gas temperature, of H2O and CO2 at two different times (0.26s and 1.08s) showing the evolution of the phenomenon. At time 1, an energy convective-conductive transfer is well observed. We also notice the presence of water vapor (H-2Omax = 0.14) and the formation of pyrolysis gases (CO2max = 0.28) while the gas temperature reaches Tmax = 1717.K) Later, at time 2, an increase in gas temperature (Tmax = 3068.K) is due to the combustion of the pyrolysis gases (CO-2max = 0.095) and the decrease of the water (H2Omax = 0.017). The propagation to the left of the flame front can be observed : the maximum values of the variables have moved towards the unburnt part of the litter. 4. Conclusions A mixed Finite Volume/Finite Element method to solve a 2D multiphase radiative and reactive model of line-fire propagation is proposed. The different physical phenomena are well observed (heating, pyrolysis and combustion). The first results obtained with this model are very encouraging. Physically, a model including soot formation will be considered to show the importance of the radiation in such combustion problems. Numerically, this kind of unsteady problems is very time consuming. An Adapted accurate implicit solver is in progress. Aknowledgment The authors thank M. Larini and B. Porterie for having provided the physical model and the set of equations. This work has been financed by the European Economic Commission under the contract EFAISTOS No.ENV4-CT96 0299. 5. Bibliography [LFE 85]
FEZOUI, L., Resolution des equations d'Euler par un schema de van Leer en elements finis, Rapport de Recherche INRIA No.358, Janvier 1985.
[LFE 89]
FEZOUI L. et al., Resolution numerique des equation de Navier-Stokes pour un fluide compressible en maillage triangulaire, Rapport de recherche INRIA No. 1033, mai 1989.
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[GRI 85]
GRISHIN A.M. et al., Study of the structure and limits of propagation of the front of an upstream forest fire, translated from : Fizika Goreniya i Vzryva, 21(1), pp.11-21, 1985.
[LAR 99]
LARINI M. et al., A multiphase formulation for fire propagation in heterogeneous combustible media, Int. J., of Heat and Mass Transfer (to be published).
[BLA 89]
LARROUTUROU, B. et al, On the equations of multicomponent perfect or real inviscid flow, Carasso, Charrier, Hanouzet et Joly Editors, Non linear hyperbolic problems, pp.69-98, Springer-Verlag Heidelberg, 1989.
[FOR 98]
PORTERIE B. et al., Wildfire propagation : a two-dimensional multiphase approach, Combustion, Explosion and Shock waves, Vol.34, No.2, 1998 also in Fizika Goreniya i Vzryva, Vol.34, No.2, 1998.
[ROE 81]
ROE, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J.C.P., 43, pp.357, 1981.
[SAK 96]
SAKAMI M. et al., Application de la Methode des Ordonnees Discretes au Transfert Radiatif dans un Milieu Bidimensionnel Gris a Geometrie Complexe, Rev. Gen. Therm., Vol.35, pp.83s-94s, 1996.
[SCH 99]
SCHALL E. et al., A mixed Finite Volume/Finite Element method applied to combustion in multiphase medium, INRIA Report, to appear.
[VLE 72]
VAN LEER B., Towards the ultimate conservative difference scheme I: the quest of monotonicity, Lecture notes in Physics, Vol.18, pp.163, 1972.
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Figure 1: Isovalues of the temperature - time 1 and time 2
Figure 2: Isovalues of H2O - time 1 and time 2
Figure 3: Isovalues of CO? - time 1 and time 2
Turbulence Modeling for Separated Flows
L.J. Lenke and H. Simon Department of Turbomachinery, University of Duisburg D-47048 Duisburg
The influence of the turbulence modeling on viscous flow field calculations has often been discussed in the past. For a meaningful comparison of different turbulence models the access to reliable measurement data is necessary. Therefore, the 2-D flow through a symmetrical diffuser with a large asymmetrical separation is chosen to investigate the separation behavior of three different two-equation turbulence models. The k-uj-, a low-Reynolds-number k-e- and an explicit algebraic Reynolds stress model are considered in this investigation. The numerical results of the algebraic Reynolds stress model are similar to the k-cj-model in prediction of the reattachment length but are in better agreement with experimental data downstream the reattachment. Significantly improved numerical results were obtained compared to the k-e-model. Furthermore, different 3-D calculations of the flow within the return channel of a multi-stage centrifugal compressor are presented. Especially at off-design conditions the turbulence models deviate notably in their prediction of separation and show great differences in the calculation of the 3-D flow structure and secondary flows. Key Words: k-cj, k-e, algebraic Reynolds stress model, separation, return channel
diffuser,
1. Introduction There are already a large number of two-equation turbulence models but a basic problem of these kind of models, namely, their failure to correctly predict the amount of separation in adverse pressure gradient flows, is still unresolved. Among the various models in existence, the fc-e-model is currently most popular and applicable to many practical complex flows with reasonable computational
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economy and accuracy. However, for the prediction of large flow separations and recirculations especially with regard to the prediction of the flow downstream of the separation the k-e- and fc-w-model have to be abandoned in favor of the higher order turbulence models which will increase strongly the numerical effort. Another point of criticism is that the fc-e-models need damping functions within the boundary layer. Recently, the explicit algebraic Reynolds stress models (ARS) which are applied in the present paper within the context of the fc-e-equation extend the range of applicability of two-equation models. This will be illustrated by two test cases (2-D diffuser and return channel) involving separations at plane and curved surfaces. The 2-D flow through the diffuser is characterized by a large asymmetrical separation at one side of the diffuser. The second test case show the 3-D flow within a return channel which is typically to join the exit from one stage of a centrifugal machine to the inlet of the next stage. 2. The Numerical Scheme
In this investigation the code developed by Reichert [REI 95] with a finite volume formulation of the full Navier-Stokes equations is used picking up elements from Roe's and Osher's scheme. Furthermore, only steady state solutions are considered. For high convergence rates, an implicit Newton-Raphson-like iterative method is used to solve first the averaged conservation equations and then the modeled transport equations but with different CFL numbers for each set of equations. Furthermore, the local time step size is calculated using a CFL number, which is a function of the local change of the density (details are described in Lenke and Simon [LEN 97a]). For the simulation of turbulent flows non linear eddy-viscosity models are an increasingly popular approach, motivated principally by the desire to combine the physical realism offered by second-moment closure with the simplicity and numerical robustness of linear eddy-viscosity models. Gatski and Speziale [GAT 93] derived an explicit algebraic stress equation for three-dimensional turbulent flows which must be solved in conjunction with two transport equations. In the present study the algebraic stress equation is solved with the standard fc-equation and an e-equation which is extended by an additional production range time scale and a cross-diffusion term (ARS-model) to improve the separation behavior of the turbulence model (details are described in Lenke and Simon [LEN 97b]). This ARS-model will be compared with the low-Reynolds-number fc-e-model devised by Lam and Bremhorst [LAM 81] and the fc-u-model devised by Wilcox [WIL 91]. 3. Boundary Conditions
At the inlet boundary, in both test cases the averaged total pressure and
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total density were specified. Furthermore, the distributions of the velocity components were taken from the measurements. At the outlet, in both cases a variation of a constant static pressure was used to establish the correct mass flow. All solid surfaces were modeled as rigid, non-slip and adiabatic. The other flow quantities were extrapolated from the interior. In both cases no values about the measured turbulent kinetic energy at the inlet are available so that k and e were extrapolated from the interior which leads to an averaged turbulence intensity of nearly 1%.
4. Numerical Results
4.1. 2-D
Diffuser
The performance of the two-equation turbulence models were evaluated for the flow through a 2-D symmetrical diffuser with a generating angle of 20 = 14.25°. A 118x65 grid was used and the numerical results were compared with measurements taken from Gersten et al. [GER 87]. The flow through the diffuser is characterized by a large asymmetrical separation at one side of the diffuser which is shown in Fig. la. All the turbulence models calculated such a separation but there are great differences in the prediction of the reattachment point or velocity profiles. Fig. Ib shows the velocity distributions near the beginning of the diffuser. At this position the ARS- and fc-u;-model calculate a very small backward flow which is similar to the measurements. The k-emodel calculates a symmetric velocity distribution which shows no similarity with the measurements. Fig. Ic and Id show the velocity distributions within the separation and demonstrate the underprediction of flow separation with the fc-e-model in this case. The two other models are similar and differ only within the separation. At this location the ARS-model calculates the highest negative velocities and agrees very well with the measurements. Near the reattachment and especially downstream the reattachment the results demonstrate the superiority of the ARS-model in relation to the standard two-equation models. The fc-w-model overpredicts the reattachment length and calculates downstream the reattachment great differences between the measured and predicted velocity distribution (Fig. Id and e). The ARS-model calculates the reattachment nearly correct with the highest negative velocities within the separation and the highest positive velocities downstream of the separation. 4.2. Return Channel The calculations to be presented have been done for the flow through a return channel with a small flow coefficient and a design flow angle 03 — 26° at the inlet. The geometry of the channel and the computational H-grid are
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shown in Fig. 2. The grid is a single block grid with y+ < 1. Only near the leading edge y+ has maximum values of 5 in a very small region. Two streamlines illustrate the flow at design point. The geometry of the channel is characterized by a constant channel width between point 3 and 4 with an increasing cross-sectional area up to point 4' and a decreasing cross-sectional area up to point 6. The Mach number distribution within the channel at design point is shown in Fig. 3. Due to the high circumferential component of the velocity at the inlet and the small channel width, the velocity distribution at the exit of the 180°-bend is uniform between hub and shroud. The secondary flow has no great influence on the flow. The differences between the turbulence models are small so Fig. 3 show only the Mach number distribution of the ARS-model. Each turbulence model calculate a large separation at the suction side which increases from hub to shroud and reattaches in the rear half of the vane. The models differ only by the prediction of the separation length with the smallest separation calculated by the ARS-model. This behavior is in contrast to the results of the 2-D diffuser which demonstrated the fact that the fc-e-model usually underestimates large separations. With higher mass flow rates (0:3 = 33°) the differences between the turbulence models increase. Due to the incidence angle the ARS- and fc-w-model calculate a separation near the leading edge at pressure side (Fig. 4). The k-cmodel calculates this separation too small with a higher pressure distribution in this region (Fig. 6). In the middle of the suction side the separation behavior of the turbulence models changes. The ARS-model calculates at suction side a smoother pressure distribution which agrees better with the measurements. This results by a smaller separation at mid span compared to the k-eand fc-cu-model. In contrast to the ARS-model the fc-cj-model calculates always larger separations compared to the fc-e-model which agree better with the measurements at pressure but not at suction side. Within the 180°-bend and in the range of the leading edge the Mach number distributions of the turbulence models are very similar. But downstream of the separations the ARS-model calculates relatively high values of the turbulent viscosity within the whole channel. This leads to a very homogeneous Mach number distribution between the rear half of the vanes and within the 90°bend. The k-e- and fc-w-model calculate in wide areas no turbulent viscosity like a laminar flow with more vorticity especially within the 90°-bend (Fig. 5). But the great differences between the Mach number distributions have no significant influence on the static pressure distribution. 5. Conclusions An explicit algebraic Reynolds stress model has been tested against a low Reynolds number k-e- and fc-w-model for a diffuser flow which involves a large separation. The comparisons with experimental data show that the ARS-model
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does provide significant improvements over the standard two-equation models especially with regard to prediction of flow separations. For the flow through a return channel which is characterized by strong stream line curvature the comparison of the turbulence models shows minor improvements of the ARSmodel. These improvements mean not only the enlargement of separation at pressure side like the k-u-mode\ but also the reduction of separation at suction side. Furthermore, the ARS-formulation used in the framework of fc-e-equations is as robust as fc-e-models. Thus, these kind of turbulence models increase the range of applicability of two-equation turbulent closure formulation at a minimal increase in cost and storage. 6. Bibliography [GER 87]
Gersten, K., Hartl, A., Pagendarm, H.G., Optimierung von Diffusoren bezuglich der Diffusorstromung und der Diffusorwande. VDI-Verlag Dtisseldorf 1987.
[GAT 93]
Gatski, T.B., Speziale, C.G., On explicit algebraic stress models for complex turbulent flows. Journal of Fluid Mechanics 254 (1993) 59-78.
[LAM 81]
Lam, C.K.G., Bremhorst, K.A., Modified Form of the ke-Model for Predicting Wall Turbulence. Journal of Fluids Engineering 103 (1981) 456-460.
[LEN 97a]
Lenke, L. J., Simon. H., Viscous Flow Field Computations for a Transonic Axial-Flow Compressor Blade Using Different Turbulence Models. ASME Paper 97-GT-207 (1997).
[LEN 97b]
Lenke, L. J., Simon. H., An Improved Algebraic Reynolds Stress Model for Predicting Separated Flows. Beijing, Proceedings of the 7th International Symposium on Computational Fluid Dynamics, 15.-19. September 1997.
[REI 95]
Reichert, A.W., Stromungssimulationen zur optimierten Gestaltung von Turbomaschinenkomponenten. Dissertation, Duisburg, Hansel-Hohenhausen Verlag 1995.
[ROT 84]
Rothstein, E., Experimented und theoretische Untersuchung der Stromungsvorgange in Rilckfuhrkanalen von Radialverdichterstufen, insbesondere solcher mit +geringen Kanalbreiten. Dissertation, Aachen 1984.
[WIL 91]
Wilcox, D.C., Comparison of Two-Equation Turbulence Models for Boundary Layers with Pressure Gradient. AIAA Journal 31 (1993) 1414-1421.
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Figure 1: Flow through a 2-D diffuser with 20 = 14.25° (a) streamlines with the ARS-model; b)-e) velocity distributions)
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Figure 2: Streamlines and grid of the return channel (44^-37x162 points)
Figure 3: Mach number distribution at design point (0:3 = 26°).
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Figure 4: Velocity and Mach number distributions within the return channel (ARS-model; a3 = 33°).
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Figure 5: Mach number distributions within the return channel (k-€-model; a3 - 33°j.
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Figure 6: Surface pressure distributions at mid span (a$ = 33°).
Simulation of unsteady Flow in a Vortex-Shedding Flowmeter
Stephan Perpeet, Andreas Zachcial and Ernst von Lavante Institute of Turbomachinery, University of Essen, D-45127 Essen, Germany
ABSTRACT Most commercial vortex-shedding flowmeters rely on known relationship between the vortex-shedding frequency and the mass flow, needing regular and well defined vortex structure as well as shedding mechanism. However, it has been observed that some designs result in rather irregular pressure signature of the vortex system, leading to problems in the signal processing. In the present study, the flow about the bluff body in a vortex-shedding flowmeter was numerically investigated using a solver of the unsteady, compressible Navier-Stokes equations in two and three dimensions. The computations were compared with experimental results obtained by ultrasonic measurements downstream of the bluff body. Several different body shapes were studied, trying to optimize the resulting pressure signature downstream of the body. Furthermore, first results of the influence of pulsations on the flow will be presented. Key Words: Vortex-shedding flowmeter, Low Mach number, Pulsation
1. Introduction Many aerodynamic problems require a volume- or mass-flow data for its quantitative solution. Therefore, a number of methods for flow rate measurement have been developed. In the present work, attention was paid to the so-called vortex-shedding flowmeters. Commercial flowmeters use a large variety of bluff body shapes, often restricted by the attachment of the pressure sensors or, more likely, the patent laws. In principle, the vortex-shedding flowmeters use the separation frequency of vortices behind a bluff body to measure the mean flow velocity of a fluid flow (Figure 1). Downstream of the bluff bo-
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dy, a von Karman vortex street develops; it's width and distance between the vortices depends on this frequency, and therefore on the bluff body's shape. Preferably, the vortex-shedding frequency should depend linearly on the mean flow velocity for a wide Reynolds number range. The dependency of the vortex frequency /, the mean flow velocity um and the width of the bluff body d is expressed by the dimensionless Strouhal number:
Figure 1. Principle of a vortex-shedding flowmeter Previous simulations of some current bluff body designs lead to fairly irregular pressure signatures, making them unreliable. It was, therefore, decided to investigate a few alternate bluff body designs. In the presently used ultrasonic method, the bluff body shape is restricted only by the required minimum strength and rigidity to avoid vibrations. The design could be optimized regarding the pressure signature or the pressure drop across the body. The signal processing requires well-defined vortices at only one dominant frequency, without any secondary effects. Therefore, an economic signal processing is only feasible with an optimized shape of the vortex body. 2. Numerical Algorithm
The numerical algorithm employed uses the three-dimensional, time-dependent full Navier-Stokes equations describing the conservation of mass, momentum and energy of the flow. The divergence form in body-fitted, curvilinear coordinates is:
with Q — J~l(p pu pv pw e)T the vector of the conserved variables. J is the Jacobian of the coordinates transformation from physical ( x , y , z , t ) to computational (£,r7,£,r) space. The program is based on the finite-volume formulation, using a cell-centered organization of the control-volumes. The spatial discretization is carried out with the help of Roe's Flux Difference Scheme,
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a Godunov-type method providing an approximate solution of the Riemann problem on the cell interfaces. Here, the flux is [VAT 87]:
The index i + | describes the values on the cell interfaces, R and L the values right and left from it. The Roe-averaged matrix A is given by the differentiation of the local-linearized function F(Q). This scheme is formally a central difference type plus a damping term. The method has been proved to be very accurate and effective in the simulation of low Mach number viscous flows [LAV 93]. Upwind-biased differences are used for the convective terms, central differences for the viscous fluxes. Starting with a constant initialization of the scalar variables and body-fitted velocity components, the integration in time is carried out by a modified explicit Runge-Kutta time stepping as well as, optionally, an implicit Approximate-Factorization method (AF) or SymmetricGauss-Seidel (SGS) scheme. 2.1. Low Mach Number Modifications Numerical algorithms for the simulation of compressible flows become inefficient and inaccurate at very low Mach numbers. The difficulties are due to the formulation of the governing equations in their discretized form. The problems have been addressed by many previous investigators, including Shuen et al. [SHU 93], Fletcher et al. [CHE 93] and Edwards et al. [ROY 98]. The main problem is the stiffness of the governing equations at low Mach numbers. The condition number of the Jacobian matrices (ratio of the maximal to the minimal eigenvalues) increases to infinity as the Mach number approaches zero. The time marching step therefore is restricted because of the large disparity of the eigenvalues of the Jacobian, representing the convective and acoustic signal speeds, respectively. The second problem results from the pressure singularity in the momentum equations at very low Mach numbers. The ratio of the magnitudes among the pressure term p and the convective terms pu2 is inversely proportional to the Mach number squared M2. The large difference in magnitude will yield a large roundoff error. To enable efficient numerical solutions of the equation system at low Mach number, a pseudo-time term is added to the time-dependent compressible Navier-Stokes equations. The primitive variables are employed as unknowns, rather than the traditional conservative variables. A preconditioning matrix F is used in order to eliminate the time-step difference between the convective and acoustic characteristic speeds at low Mach number. The preconditioned
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equations can be discretized in the delta form:
where the Jacobian matrices are defined as,
Here Q is the vector of the conservative variables and Q is the vector of the primitive variables (p, u, v,w, T)T. A more detailed description of the matrix F can be found in [YAO 97]. To circumvent the problem of large difference in magnitude of the convective and pressure terms in the momentum equations, a gradient splitting of the Euler flux into convective terms and a pressure term can be considered: Both split gradients are of the same order of magnitude, independent of the Mach number. The Liou's Advection Upwind Splitting Method is applied to treat the convective and pressure terms separately. The convective terms are upstream-biased using an appropriately defined advection Mach number at the cell interface, while the pressure term is strictly dealt with by using acoustic waves. One of the advantages of the method is that the upwind effect can be easily reached with few modifications of the programs. 2.2. Verification The present numerical algorithm was subjected to verification of it's temporal and spatial accuracy and consistancy. The scheme is formally second order accurate in space, since the viscous terms are obtained from second order central differences. The scheme was first verified using the usual grid refinement study for the case of viscous flat plate flow at a free stream Mach number of MOO = 0.5. Defining the global error as the I/2 - norm of the deviation of the present solution from the Blasius solution, second order accuracy was verified (von Lavante [LAV 90]). Next, the combination of the present solution scheme with the computational grid was investigated for the case of the "new design" bluff body (see below) using three different grids. Table 1 summarizes the results. It should be noted that the finest grid very closely approaches the Strouhal number of the corresponding experiment.
Fields of application Meshpoints
A* [a] rel. CPU-time
f(Hz] Strouhal number
5000 8.8 • 10-7 1 342 0.267
12300 5.88 • 10~7 10 339 0.265
433
41600 || exp. 7 2.06 -KT 35 335 0.262 0.263
Table 1. Comparison of different grids The temporal accuracy was tested by simulating the 2-D flow about a cylinder, with free stream Mach number M^ = 0.1 and a Reynolds number of Re = 200. The resulting Strouhal number of the vortex separation is compared with known experimental and numerical results in Table 2. The agreement is also very good. Lugt Truckenbrodt Authors Faden
Method experimental experimental numerical numerical
Strouhal num. 0.193 0.192 0.194 0.198
Table 2. Strouhal number in cylinder wake flow 3. Results In the present research project, several shapes of the bluff body were investigated experimentally and numerically. The experimental results have been given in detail by Hans et. al. [HAN 98]. Main emphasis was put on linear dependency of the vortex-shedding frequency as a function of the mean flow velocity. In the present paper, only the three most interesting bluff body shapes, shown in Figure 2, will be discussed. Two of the bodies, a "T"-bar and a rectangle, were fairly simple, the third one was optimized by the present authors for the cleanest signal and was therefore called "new design".
Figure 2. Design of the bluff body shapes The horizontal part of the T-bar was facing upstream, which was in contrast to other applications of this shape [MIA 93]. The Strouhal number (Figure 3) was constant for the whole velocity range corresponding to Reynolds numbers
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from 10,000 to 300,000. Experimental and numerical results showed a very good agreement. The standard deviation was within 2%.
Figure 3. Strouhal number versus Reynolds number Although the two-dimensional simulations predicted vortex-shedding frequencies that agreed very well with the experimental data, it was decided to investigate the flow about the promissing "new design" using three dimensional approach. No symmetry was assumed, allowing the specification of asymmetric inflow velocity profiles. The dominant vortex structure makes this case interesting for large eddy simulation (LES) turbulence treatment. Therefore, no turbulence modell was used, although the inflow would correspond to a fully developed turbulent pipe flow. The computational grid consisted of 22 blocks. Two grid resolutions were used: coarse with 300, 000 cells and fine with 1.07 • 106 cells. The coarse grid was implemented in order to study the process of the main vortex generation. No attempt was made in this case to properly resolve the boundary layer at the pipe wall. There were only a few grid points in this boundary layer. The second, finer grid consisted of 1.07 • 106 points, making the full simulation of all the viscous layers possible. The minimum grid spacing at the wall was less than y+ = 1.0, and of the same order of magnitude as the Kolmogorov length. There were at least 20-40 grid points in each of the shear layers. The grid was exponentially stretched toward regions with smaller gradients of the flow variables. The resulting flow displayed highly three-dimensional nature, with main vortices curved toward the walls, and with a complex system of secondary flow features. After separating from the bluff body, the primary vortices, seen in Figure 4, were convected downstream. They dissipated toward the pipe walls; their central part became straight, assuming almost two-dimensional form. In order to visualize some of the lower order features, such as secondary vortex structure, the enstrophy, defined as uj — \ (V x U)2 was computed and displayed in contour plots. The enstrophy contours on a cylindrical surface close to the pipe wall are shown in Figure 5. Here, the so called "horse shoe vortex" is clearly visible in front of the bluff
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body. Behind it, the highly vortical, unsteady flow associated with the primary vortices can be seen. Several secondary and tertiary vortices, some aligned with direction normal to the main vortices, could be observed, indicating that at least some of the turbulent structure was simulated.
Figure 4. Density contours over the pipe cross-section
Figure 5. Enstrophy contours at the wall
Figure 6. Pressure evolution for pulsation frequency of lOOHz
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Finally, the effects of pulsating inflow on the function of the vortex-shedding flowmeter are currently investigated. As can be seen in Figure 6, pulsations in the flow with frequencies close to the natural vortex-shedding frequency had an pronounced influence on the operation of the meter. Between two pulsation peaks the former shedding frequency recovers. 4. Conclusions
A verified and validated Navier-Stokes solver for two-dimensional and threedimensional simulations of compressible, viscous and unsteady flow over a wide range of Mach numbers is presented. The solutions achieved by this program show good agreement with experimental data. The effects of disturbed inflow conditions are currently investigated. 5. Acknowledgement
This project is supported by the Deutsche Forschungsgemeinschaft (DFG).
6. References [ROY 98] [HAN 98]
[MIA 93]
[CHE 93] [SHU 93] [VAT 87]
[LAV 90] [LAV 93]
[YAO 97]
J. R. EDWARDS AND C. J. ROY, AIAA Journal, Vol. 36, No. 2, pp. 185-192, 1998. HANS, V., POPPEN, G., LAVANTE, E. v., PERPEET, S.: Vortexshedding flowmeters and ultrasound detection: signal processing and bluff body geometry. Flow Measurement and Instrumentation 9 (1998), 79-82. MIAU, J.J., YANG, C.C., CHOU, J.H. AND LEE, K.R., A T-shaped vortex shedder for a vortex flowmeter, Flow Meas. and Instr., Vol. 4, No. 4, 259 - 267, 1993. R. H. PLETCHER AND K.-H. CHEN, AIAA-93-3368-CP. J.-S. SHUEN, K.-H. CHEN, AND Y. CHOI, J. of Comp. Phy. 106, pp. 306-318, 1993. VATSA,V.N.,THOMAS,J.L.,WEDAN,B.W., Navier-Stokes computations of prolate spheroids at angle of attack, AIAA Paper, (87-2627), 1987. VON LAVANTE, E., The Accuracy of Upwind Schemes Applied to the Navier-Stokes Equations, AIAA Journal, Vol. 28, No. 7, 1990. VON LAVANTE, E., YAO, J., 1993. Simulation of flow in exhaust manifold of an reciprocating engine. AIAA 24th Fluid Dynamics Conference. J. YAO AND E. VON LAVANTE, Proceedings of 7th International Symposium on CFD, pp. 689-694, Sept. 15-19, 1997, Beijing, PR China.
A Finite Volume Scheme for the Two-Scale Mathematical Modelling of TiC Ignition Process
A. Aoufi LIMHP-CNRSs Avenue JB Clement 93430 Villetaneuse
V. Rosenband Faculty of Aerospace Engineering Techmon Haifa-32000
ABSTRACT We consider the problem of the ignition of a cylindrical sample of infinite length composed of Ti particles coated with a thin TiC layer and placed into a furnace at T. We describe an implicit finite-volume scheme, for the two spatial scales modelling, on a moving mesh at the particle level, applied to the numerical study of the ignition process. Key Words: Reaction-Diffusion Equation, Adaptive Finite- Volume, Stefan Problem, Moving-Mesh, Ignition Process, Solid-Solid Combustion.
1. Introduction
In this paper we present a two-spatial scales mathematical modelling for Ti + C ->• TiC ignition process [AOU 98, LAP 96, SEP 95]. At the particle level, we have a Stefan problem which is discretized by a finite-volume scheme on a moving mesh. The paper will be organized as follows, in section 2 we describe the mathematical modelling, in section 3 we present the numerical scheme we have applied to the treatment of this two-scale , and describe the finite volume scheme. Section 4 presents some numerical simulations.
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2. Mathematical Modelling The reader is referred to [AOU 98] for more explanations about the physical analysis of the process.
Figure 1: Scheme of the reaction modelling
2.1 Mass Balance We consider the mass-diffusion balance expressing Pick's law for a spherical particule of Titanium of radius Rp. We denote by Q = f^2 (<)UQi (t] = [Q,R(t)] U [R (t) , Rp] = [0, Rp] .The mass-diffusion coefficients follow an Arrhenius-type law. 2.2.1 Dissolution Domain , Zone II
2.2.2 Carbidization Domain , Zone I
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2.2 Stefan Problem We express the condition at the interface which indicates that due to concentration gradient in both domains I and II, the interface moves with the velocity
2.3 Energy Balance We evaluate the overall energy balance at the sample macroscopic level, for which we take into account the heat released at the mesoscopic scale - i.e. particle scale- by the exothermic mass diffusion processes described below. 2.3.1 Carbidization Domain , Zone I We denote by j\ ' the rate of heat released during carbon formation - by diffusion process in zone I-, denned by
2.3.2 Dissolution Domain , Zone II 11
We denote by dO^dt\
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2.3.3 Global Heat Balance We consider the cylinder domain £lc = [0, RC] and write for V z G [0, RC] The heat balance takes into account the heat released locally at the particle level for the two diffusion processes by:
2.3.4 Boundary Conditions Symmetry Condition at the center of the cylinder
Heat supplied by the furnace maintained at temperature T^
3. Numerical Scheme
We integrate from tn to tn+\, the three different balances, assuming that a non uniform temporal mesh is given. We use a variable stepsize backward difference formula of order 1, which is well suited for the numerical treatment of stiff equations. We focuss on the numerical methodology used to discretise the mass balances, written in conservation form, on a variable domain. In order to track efficiently the variation of the computational domains QI (t) and ^2 (t) , we have considered the integration of the mass balance on a cell K i ( t ] = [ n ( 0 > r » + i ( 0 ] wmch evolves with time and denotes its boundary dKi (t) = {ri (t) , r,-+i (t)} . We adopt a cell-centered formulation. 3.1 Integration on a moving mesh
The transport theorem of Reynolds [N'KO 94] affirms that:
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where t>;+1 (t) = r 'dt ' and i;,- (t) = r^ ' are the velocities of the extremities of the intervall on which the mass balance is integrated. We outline the procedure, and omit the details of the algebraic computations. We integrate the previous relation from tn to tn+\ and substitute ^ ( z , r , t ) from its definition into the mass balance. We apply Green's Formula for the discretization of the divergence of the mass fluxes. We apply the first order accurate backward Euler scheme for the discretization of the right hand side integrals. We denote Atn = volume of the cell and end up with,
We point out that the following term, which takes into account the transport of the concentration field, due to the variation of volume of the integration cell appears :
Since C is constant in cell KI (tn) , we may use "local information" i.e. related to this cell, and therefore write the following continuity relation
We now have the following relationship
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After some algebra and leaving the convected mesh terms in the right hand side, we are led to a three diagonally dominant matrix system which can be efficiently inverted by the classical -double sweep- Thomas Algorithm. At this stage we have made no assumption on the nodes velocities. 3.2 Nodes Velocity We have to prescribe the speed of the mesh; this is realized through a linear speed formula, according to [MUR 59], taking into account the fact that the velocity at the interface between domain I and II is known from the stefan problem equation. 3.3 Boundary Conditions An important issue is the discretization of the boundary conditions on a moving mesh, because they will lead to a different set of equations that will influence significantly the behaviour of the numerical solution, if a wrong formula is supplied, for example consider: Constant We integrate the equation on cell Kj (tn) = assuming that therefore and the equation is,
Using the previous arguments on the continuity of the concentration field we get
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In fact the variation of the computational volume can be understood as a source term. 3.4 Time Step Adaptation The stiffness of the problem means that once the heat released by the chemical kinetics is enough, a sharp rise of the temperature occurs inside the sample from the exterior surface to the interior of the cylinder. A time-stepping adaption procedure to dynamically reduce the time-step with respect to the temperature field behaviour has been considered. 5. Numerical simulations
Ignition temperature is reached, when a sharp increase in temperature field occurs. This ignition process depends on particle size, initial thickness of TiC interfacial layer, heating rate, and value of TOO. We present below a sample of such temperature profile.
Figure 2: Temperature profile T(Rc,t)
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[AOU 98] . AOUFI A., ROSENBAND V., Mathematical modelling and numerical simulation of Titanium/Carbon Ignition , 2nd International High-Energy Materials Conference and Exhibit, Dec 8-10 (1998). [LAP 96]
. LAPSHIN O.V. AND OVCHARENKO A.E. , A mathematical model of high-temperature synthesis of the intermetallic compound Ni^Al during ignition , Combustion, Explosion and Shick Waves , Vol. 32, N° 2,(1996),158-164.
[MUR 59] MURRAY W.D. AND LANDIS F., Numerical and Machine Solutions of Transient Heat-Conduction Problems Involving Melting or Freezing , Transactions of the ASME, (May 1959), 106-112. [N'KO 94] N'KoNGA B. AND GUILLARD H., Godounovtype methods on nonstructured meshes for three-dimensional moving boundary problems , Comput. Meths. Appl. Mech. Engrg. 113 (1994). [SEP 95]
SEPLYARSKII B.S. AND GORDOPOLOVA I.S., Ignition of condensed systems interacting through a layer of high-melting products , Combustion, Explosion and Shock-Waves, Vol. 31, N°4, (1995), 405-410.
Two Perturbation Methods to Upwind the Jacobian Matrix of Two-Fluid Flow Models.
Kumbaro A., Toumi I. CEA Saclay, DRN/DMT/SYSCO F-91191 Gif-sur-Yvette Cedex, France [email protected] Cortes J. CEA Cadarache, DRN/DTP/SMET F-13108 Saint-Paul lez Durance, France
ABSTRACT We examine the eigenstructure of a two-fluid model Because of complex interphase interactions, perturbation methods are used to get approximations of the eigenelements. We compare two significant perturbation methods. Mathematical and numerical results are provided. Key Words: two-phase flows, eigenstructure, upwinding, perturbation methods.
1. Introduction In this paper, we examine the eigenstructure of a two-fluid model, derived from [TOU 96], when regularising terms are added to it. We think that such a study is of relevance to construct upwind schemes. Unfortunatly, it turns out to be a difficult task in two-fluid systems. Hence, several approaches are proposed to get practical approximations of the eigenelements : • A numerical algorithm. However, it may be expensive in CPU time for 3D finite volume calculations, although improvments are in progress ([ALO 98]). Also intensive use of symbolic calculation packages may help obtain closed form related expressions.
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• A perturbation method using a small parameter. We distinguish : — the density perturbation method to analyse the eigenstructure when the density ratio is small ( c = ^ < 1), see [COR& 98],[COR 98]. — Taylor expansion of the eigenvalues when the relative velocity is small compared to the speed of sound ( f = ^-
Two-fluid models are regularised by the addition of interface pressure correction terms [POK 97]. Indeed, in the basic model, surface tension effects are neglected, and it is assumed that pressure equilibrium exists between the two phases : p = pi — pg ^ Pi, with pi the interface pressure. Here, we give the standard homogeneous form of the physical set of conservation equations [TOU 96] :
where the subscript k refer to the gas phase ( k = g ) or to the liquid phase ( k — 1). otk is the volume fraction ( ag +&i = 1 ), pk the density, Vk the velocity, 6k the energy and h^ the enthalpy ( hk = &k + ^ )• Moreover, we note by Hk and Ek respectively the phasic total enthalpy and energy. For simplicity, we will consider a perfect gas. Several interface pressure correction models exist in the litterature. They must vanish when the velocities of the two phases become identical (vr = 0 ). We will apply the two perturbation methods to : • the pressure correction proposed in [LAH 92]
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• the pressure correction proposed in [TOU 96]
Actually, setting
we see that Cp = ag8 leads to correction (3) whereas 6 = 0 and Cp =• Cp(ag) leads to correction (2). We can compute the Jacobian matrix A of the two-fluid system :
As mentioned in the introduction, the interphase interactions yield a complex eigenstructure. Hence, perturbation methods may be a convenient way to get practical approximations of the eigenvalues and eigenvectors. 3. The two perturbation methods
3.1. The Taylor expansion of the eigenvalues In order to determine the eigenvalues of the system, we must find the roots of a polynomial of degree six P(A) given by :
A straightforward computation of this determinant leads to the following polynomial :
To get approximations of the roots of PA , one can use a perturbation method by introducing the ratio
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and the new variable sound in the two-phase mixture given by
with the 'characteristic' speed of
We look for a first order approximation of the eigenvalues and the eigenvectors (see [KUM 96]). Let the polynomial PA (A) be writen as
with We look for the roots of jP(z;£) in a neighbourhood of a root ZQ of the polynomial p o ( z ) . We will carefully distinguish the single roots and the double roots of Po(z}. Hence, introducing the notation 60 — 7 ^'^2 , we can find first related approximations of the two-fluid eigenvalues :
where k = I corresponds to the pressure correction (2) and k — ag correspond to the correction (3). Following the same method, the first order approximation of the eigenvectors can be derived (see [TOU 96]).
3.2. The density perturbation method In [COR& 98], it is shown that a scaling of the densities in the two-fluid system yields a splitting of the Jacobian matrix :
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t is the average density ratio e = p° /'p° and
7-A-i + AQ would represent the Jacobian matrix of a two-fluid system with no pressure gradient in the liquid phase. Hence, it is expected that its eigenelements are easy to compute. For instance, we get
The computations yield :
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In particular, when A-\ — 0, the perturbation theory for linear operators provides a convenient way to obtain higher approximations of the eigenelements of the original matrix ([HIN 91]). For instance the correction (2) yields A-i = 0 and in these circumstances, we can expand the eigenelements into a first order approximation in e ( [COR& 98]).
4. Numerical results The presence of non-conservative products yield significant difficulties. Here, we mainly follow [KUM 96]. Hence we construct the Roe type numerical scheme as follows :
with the numerical flux given by
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The positive and negative part of the Roe-averaged matrix are obtained thanks to first order approximations of the eigenstructure above. Some two-phase shock tube tests are computed using the two perturbation methods described in 3 and the two pressure corrections mentioned in 2. Moreover, we take 7 = 1.093 and pi = 720. The initial conditions for these shock tube problems are : TEST 1 Left state Right state
0.25 0.10
TEST 2 Left state Right state
ag 0.8 0.8
Oig
vg (m/s) 0 0 vg (m/s) 150 100
vi (m/s) 1 1 vi (m/s) 5 1
p (MPa) hg (kJ/kg) 3092.7 20 3099.9 10 p (MPa) 0.15 0.1
hi (kJ/kg) 1338.2 1343.4
hg (kJ/kg)
hi (kJ/kg)
12 12
3 3
The CFL number is 0.9 and the computations have been done with 400 cells. Results are provided after 200 iterations. The theorical results obtained in the previous section and the numerical results obtained in this one will be fully discussed in the presentation.
[ALO 98] F. ALOUGES, Une methode iterative pour decentrer sans diagonaliser, private communication [COR& 98] J. CORTES, A. DEBUSSCHE AND I. TOUMI, A Density Perturbation Method to study the Eigenstructure of Two-Phase Flow Equation Systems, Journal of Comp. Physics, Vol.147, No.2, 1998, pp. 463-484 [COR 98] J. CORTES, An Asymptotic Two-Fluid Model for Roe-Scheme Computation, Computational Fluid Dynamics ( ECCOMAS'98 proceedings ), John Wiley & Sons, Ltd., Vol.11, 1998, pp. 416-422 [EYM 99] R .EYMARD, T. GALLOUET, R. HEREIN, Finite Volumes Methods, to appear in the Handbook for Num. Anal., eds C. Lions, North Holland [GOD 91] E. GODLEWSKI, P . A. RAVIART, Hyperbolic systems of conservation laws, Mathematiques &; Applications, Ellipses, 1991 [HIN 91] E. J. HlNCH, Perturbation Methods, Cambridge Texts in App. Math., 1991 [KUM 96] I. TOUMI, A. KUMBARO, An approximate Linearized Riemann Solver for a Two-Fluid Model, Journal of Comp. Physics, Vol.124, 1996, pp. 286-300 [LAH 92] R. T. LAHEY, JR., The prediction of phase distribution and separation phenomena using two-fluid models, Boiling Heat Transfert, Elsevier Science Publishers, 1992, pp. 85-121
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[POK 97] H. POKHARNA, M. MORI AND V. H. RANSOM, Regularization of TwoPhase Flow Models : A comparison of Numerical and Differential Approaches, Journal of Comp. Physics, Vol. 134, 1997, pp.282-295 [ROE 81] P. L. ROE, Approximate Riemann solvers, parameter vectors and difference schemes, Journal of Comp. Physics, Vol.43, 1981, pp. 357-372 [TOU 96] I. TOUMI, An Upwind Numerical Method for Two-Fluid Two-Phase Flow Models, Nuclear Science and Engineering, Vol.123, 1996, pp. 147-168 [TIS 97]
I. TISELJ, S. PETELIN, Modelling of Two-Phase Flow with Second-Order Accurate Scheme, Journal of Comp. Physics, Vol.136, 1997, pp. 503-521
Figure 1: Test 1. Eigenstructure obtained by an exact calculation and the two perturbation methods
Figure 2: Test 2. Eigenstructure obtained by the two perturbation methods
Finite volumes simulations in magnetohydrodynamics
M. Hughes, L. Leboucher, V. Bojarevics, K. Pericleous, M. Cross Centre for Numerical Modelling and Process Analysis University of Greenwich London SE18 6PF — UK
ABSTRACT Magnetohydrodynamics involves the interaction of magnetic fields with fluid flows, so that simulations in this area of physics and its applications couple the resolution of Navier-Stokes equations and Maxwell equations. While most simulations in fluid dynamics are performed with finite volumes, most of the electromagnetic computations are traditionally performed with finite elements. The straight forward temptation is to solve each problem separately using one of the available finite volumes and finite element software tools, and feed back the results of one code into the other. This method can work reasonably well when the flow does not influence the magnetic field, that is to say when the flow only is influenced by the magnetic field. Then the amount of data exchange between the two codes is affordable. However, if both fluid flow and magnetic field have an influence on one another, then the data exchange and iterative computation between the two systems becomes too large. In such a case it is preferable to solve both problem within the same code and, for simplicity, with the same method, i.e. finite volumes or finite elements. While electrical engineering finite element software often gives approximate conservation of electrical quantities without serious inconvenience, fluid dynamics codes are very sensitive to the conservation of different physical quantities, especially the conservation of mass. Therefore, a good option is to solve both fluid dynamics and electromagnetic equations with the same conservative method, i.e. the finite volume method. Examples of conservative discretisations and simulations of electromagnetic systems will be given at the conference. Key Words: MHD, magnetohydrodynamics,
finite-volumes.
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1. Introduction An electrically conducting fluid of conductivity cr and velocity u interacts with a magnetic field B when induced or imposed electric currents of density j create a Lorentz force j x B within the fluid. This force may either be used to improve some casting processes or have unwanted stabilizing or destabilizing effects on liquid metal flows. The Ohm law in a moving fluid submitted to an electric potential and a time-dependent vector potential A defines the current
The electric potential may be solved from the divergence of this law, assuming Coulomb's gauge V • A = 0 and the conservation of electric charges V • j = 0
The magnetic field can be solved from the curl of Ohm's law (1), from Maxwell's equation V x B = fj,QJ and from the identity V x V x B = —V 2 B which holds for any conservative field — V • B = 0.
This is the induction equation. Finally, the Navier-Stokes equation with the Lorentz force is
where p and v are the density and the kinematic viscosity of the fluid and where the pressure p is obtained from the divergence of this equation, taking into account the conservation of mass V u = 0.
2. Imposed magnetic fields Rewriting the induction equation in dimensionless form gives exactly the same equation with the exception that the magnetic diffusivity (HQ(T)~I is replaced with the inverse of the magnetic Reynolds number Rm = poauL. In case where this number is small, corresponding either to a low electrical conductivity (^o^)" 1 ; to a small velocity or to a system of small length scale L, the magnetic field is unaffected by the motion of the fluid. Then the induction B may be computed by any mean, e.g with an electrical engineering simulation software. The same software may be used to compute the electric potential
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Figure 1: magnetohydrodynamic velocity field of the moten aluminium in an electrolysis cell. Stirring of the liquid metal by the electromagnetic forces.
and the resulting currents and Lorentz force. This force may then be added as a source term in any other computational fluid dynamics code. This is done easily with the finite-volume simulation software PHYSICA [PHY 99] where the electric currents and magnetic fields were taken from the finite-element code CADEMA [CHI 98] in order to solve the magnetohydrodynamic flow of a molten aluminium pad lying on an electrolysis cell of an aluminium production plant. Alternatively, once the magnetic and velocity fields are known, the electric potential may be solved from eq. (2) within the fluid dynamics code using the same finite volume method as for the pressure equation (5). This method is used to model another phenomenon of the same aluminium production process: the instability of the position of the interface between the aluminium pad and the electrolyte layer lying upon it. This instability is very similar to the waves at the surface of oceans and rivers apart from the fact that it is submitted to an electromagnetic perturbation coupled to the variations of the interface position. Here only the perturbation to the electric potential is computed within the fluid dynamics code [CHI 98]. The finite volume mesh is shown on figure 2. Note that the electric potential $ is shifted from the other scalar variable 77 denoting the height of the interface, in order to obtain the value of its derivatives d<&/dx and d^/dy at the positions where the electric currents and the resulting Lorentz force need to be known, i.e. at the positions of the momentum variables U and V. An example where all electromagnetic quantities are part of the fluid dynamics finite-volume code is the flow of a liquid metal in a rectangular
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Figure 2: finite volume mesh for magnetohydrodynamic system formulated in terms of the fluid variables U, V, TJ and the electric potential $. Note the position of
pipe [HUG 94] [LEB 99] submitted to a constant magnetic field. Figure 3 shows the electric currents induced by the magnetic field in the fluid moving perpendicularly to the plane of the figure. 3. Induced magnetic fields
When the magnetic Reynolds number is not small compared to 1, then the magnetic field is affected by the velocity field according to eq. (3). This
Figure 3: electric currents in the cross section of an electrically insulating duct with increasing magnetic field strength from left to right.
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Figure 4: iterative solution of Navier-Stokes and Maxwell's equations.
equation may be rewritten as a convection-diffusion equation
It appears then that the induction equation may be discretised and solved with the same finite difference scheme as the Navier-Stokes equation (4). The induction equation could however be solved by any electrical engineering finite element code. But exchanging the velocity and magnetic fields between the fluid dynamics and the electrical codes until both fields have converged may require too much time, especially if the data need to be read and written on the disk between each call to one of the two codes. Another reason for solving the induction equation with finite volumes is the conservative properties of this method. Some finite elements codes don't satisfy V • B = 0. This may not be a problem for steady electrical engineering simulations, but it becomes a serious drawback when time dependent simulations or iterative computation of fluid velocity and magnetic fields are performed. Indeed the error in V • B can amplify with time or with the iteration procedure, leading to completely wrong results or even to numerical instabilities. An example of magnetohydrodynamic channel flow where the magnetic field is convected by a large value of the fluid velocity is shown on figure 4. The Navier-Stokes and induction equations are solved iteratively with the same finite volume method. 4. Conclusion The nature of magnetohydrodynamics where all vector fields u, B and j are divergence free together with the dynamical properties of fluid flows makes the finite volume method suit particularly well to coupled electromagnetic and fluid mechanical problems. The same finite difference scheme can be applied to
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both of them, making magnetohydrodynamic codes shorter, simpler and more accurate than classical finite elements. 5. Bibliography [PHY 99]
http://physica.gre.ac.uk
[CHI 98]
CHIAMPI, M., REPETTO, M., CHECHURIN, V., KALIMOV, A., LEBOUCHER, L., 8th International IGTE Symposium on Numerical Field Calculation in Electrical Engineering, Institut fiir Grundlagen und Theorie der Elektrotechnik, Graz, Austria, September 1998.
[HUG 94]
HUGHES, M., Computational Magnetohydrodynamics, thesis, University of Greenwich, 1994.
[LEB 99]
LEBOUCHER, L., Monotone Scheme and Boundary Conditions for Finite Volume Simulation of Magnetohydrodynamic Internal Flows at High Hartmann Number, J. Comput. Phys., Vol. 150, pp.181-198, 1999.
Finite Volume Method for Large Deformation with Linear Hypoelastic Materials
K. Maneeratana and A. Ivankovic Department of Mechanical Engineering Imperial College of Science, Technology and Medicine, London, UK
ABSTRACT This paper describes a development of the finite volume method for modelling of structural problems involving geometrical non-linearities. The rate of Green strain tensor and second Piola-Kirchhoff stress tensor are used as the work conjugate pair in total and updated Lagrangian descriptions whilst the deformation rate tensor and Cauchy stress tensor with Jaumann objective stress rate are employed in the deformed Lagrangian (Eulerian) description. The law of conservation of linear momentum is discretised and resulting systems of algebraic equations are solved, using an iterative segregated procedure. The method is applied to a number of test cases and the results are shown to be in good agreement with analytical results. The accuracy, simplicity and adaptability of the method for non-linear structural problems are clearly demonstrated. Key Words: Finite Volume method, large deformation, hypoelastic materials.
\. Introduction This work concentrates on the development of the cell-centred FV method to geometrically non-linear problems involving linear hypoelastic materials. Since the method is inherently iterative, the computational effort for solving non-linearity is not expected to be vastly difficult. The law of conservation of linear momentum in the integral form is employed as the governing equation with linear hypoelastic materials as the constitutive relationships or equations of state. In the total strain or deformation theory (DT) scheme, the Green strain and second Piola-Kirchhoff stress are used. As hypoelastic constitutive equations have to express relationships between stress and strain rates,
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solids described by total strain formulation are, therefore, not hypoelastic materials. However, the St. Venant-Kirchhoff hyperelastic model, which has similar constitutive relationships to hypoelastic materials, are employed in this study [BON 97]. In the incremental approach, however, there are several possible schemes. The total Lagrangian (TL) and updated Lagrangian (UL) schemes are based on the initial and updated initial configurations, respectively. In both schemes, the rates of Green strain and second Piola-Kirchhoff stress are used as the work conjugate pairs. In the deformed Lagrangian (DL) scheme, the rate of deformation and Cauchy stress with Jaumann objective stress rate are employed. This scheme is also commonly called Eulerian [MCM 75] even though the formulations are not truly Eulerian as conventionally defined. In the incremental schemes, two methods of obtaining incremental governing and constitutive equations are considered. The derivation approach, denoted by 'deri' in the rest of the paper, simply differentiates the relationships with respect to time, while the total difference approach, denoted 'diff, considers the difference of total forces and deformations in the control volume. The governing equations are discretised by a cell-centred finite volume technique in line with earlier studies [DEM 94], [WEL98]. The presented method is, then, employed to solve simple problems and numerical results are compared with analytical solutions.
2. Governing and Constitutive Equations In this section, the governing and constitutive equations are briefly outlined. Details can be found in most continuum mechanics textbooks, such as [CRI 91] and [BAT 96]. Position of a particle within a body at a reference time t0 is defined by position vector x0. A motion, caused by applied forces, brings the particle to a new location x at a subsequent time t. In a TL formulation, all static and kinematic variables are referred to the initial configuration at time t0. Even though the formulation for DT scheme can be categorised in this approach, the abbreviation TL will be specifically referred to the incremental formulation hereafter for clarity. The UL formulation is based on the same procedure as TL scheme but variables are referred to the most recently calculated (updated) configuration. For the pure UL scheme, employed in this study, a new reference configuration xu is calculated for every increment of time or load. This updating is, in effect, resetting the displacement vectors to zero at the beginning of every increment and the TL formulations can be, therefore, simplified as will be shown in the next section. On the other hand, the deformed Lagrangian (DL) equations are based on the current deformed configurations x.
2.1. Governing
Equations
The basic law of conservation of momentum for the deformed body of volume v, bounded by deformed surface with normal vector a pointing outwards, is:
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where u is the displacement vector, CF is the Cauchy or true stress tensor, b is the body force, p is the current density and a dot above a quantity represents the time derivative of that quantity. As the Cauchy stress is based on the actual force, acting on deformed volumes, the second Piola-Kirchhoff stress tensor, S, is introduced as the mathematical mapping of the Cauchy stress to the undeformed volume as S — JF~l • o-(jP"1 )r where F = I + G is the deformation gradient tensor, G = du/dx0, I is the identity tensor and the Jacobian / = det(F). Therefore, equation [1] can be rearranged for DT approach for an undeformed body with initial surface vector a0, volume v0 and density p,, as:
where deformed surface vector a and volume v relate to a0 and v0 by da = J(F~>)7 -dao and dv = Jdvo. The equation [3] can be expressed in TL scheme using 'derivation' and 'difference' schemes as:
Equations [3] and [4] can be simplified for UL formulations to:
where the second Piola-Kirchhoff stress Su, rate of deformation gradient Fu, density pu, surface vector au and volume vu are referred to the last updated configuration. The reference position vector xu is updated every increment by setting it equal to jc of the previous increment, giving Fu = I since the tensor Fu is now based on the updated configuration. The rate form of equation [1] can be written for the DL formulation, using simple time 'derivation' and 'differencing' schemes as:
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Finite volumes for complex applications
where da - (tr(L)7 - (L) r ) • da and velocity gradient L = du/dx. If the force difference approach is used, the rate form of DL formation can be expressed by:
2.2. Constitutive
Equations
In order to obtain a well-posed system, the relationship between stress and strain measures for a given material is required. For the DT scheme, the Green strain E can be expressed using the deformation gradient F as:
The linear relationship between second Piola-Kirchhoff stress S and Green strain E can be expressed by:
where JJL and A are Lame constants. For the TL scheme, the rate form of equation [10] can be expressed as:
where the rate of Green strain obtained by simple 'differentiation' and 'different' schemes are:
For the UL scheme, the formulations of strain rate in equations [12] and [13] can be simplified to the Green strain rate in the updated configuration, Eu, as:
As UL and TL schemes are based on different configurations, constitutive relationship [11] must be corrected in order to obtain the equivalence between the TL and UL results [BAT 96]. The corrections is based on relationships between the TL and UL stress and strain rates as:
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This correction converts UL stress and strain tensors to the TL formulations but it also heavily complicates the expression. As such, there is no clear advantage of the UL formulations over the TL ones. However, if the strain is small, the equation [11] can be employed as the relationship between the rate of stress and strain measures for large deformation modelling. Therefore, this study employs this formulations for UL scheme with small strain restriction as:
Due to the change in reference configuration in UL scheme, the stress Su at the start of an increment is equal to (7 in the last increment. Therefore, the total value of Su is calculated from:
where n is the time step counter. In the DL formulation, the rate of (7 is frame different or not objective. As stress and strain measures in a constitutive relationship must be frame indifferent, an objective stress rate has to be employed. The rate of Cauchy stress a is related to the Jaumann objective stress rate G} by:
where W = |(L-L r ) is the rotation rate tensor and Z) = |(L + Lr) deformation rate.
2.3. Mathematical
is the
Model
By introducing constitutive relations into the governing equations, a general form of the transport equation is obtained in the form:
which can be regarded as an equation for the dependent variable 0, where 0 stands for displacement vector or displacement rate vector for total strain and incremental approaches, respectively, / ^ is the temporal coefficient, D $ is the diffusion coefficient, Q r
3. Discretisation Procedures The general governing equation [20] is discretised by employing a finite volume discretisation which has been successfully developed for CFD applications by
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Finite volumes for complex applications
[DEM 94] and [WEL 98]. The presented method takes the full advantage of control volumes of arbitrary shapes. It employs the fully implicit time discretisation and assumes piece-wise linear distribution of dependent variables and material properties in space. The temporal term is approximated by the mean value theorem. Surface and volume integrals are discretised using the mid-point rule. The non-orthogonality in the diffusion term is approximated by a simple orthogonal correction approach. The spatial discretisation is second order accurate while the temporal discretisation is only first order accurate. The resulting sets of coupled non-linear algebraic equations are segregated and the resulting algebraic systems of linear, de-coupled equations are iteratively solved by a pre-conditioning conjugate gradient method. 4. Test Case In order to demonstrate the method's accuracy and efficiency, few examples are modelled and numerical results are compared with analytical solutions in order to validify the code [MAN 99]. A problem presented here is a cube subjected to simple tension. The cube, with sides of length L, is discretised by 27 cubic Cvs. It is subjected to displacement V applied uniformly over one side while the opposite side is assumed to be symmetry plane. Other sides are free surfaces. The force F, required to deform the cube, is calculated from the resulting stresses on the loaded boundary. Material properties used in the calculation are E = 200 GPa and v = 0.3. Figure 1. shows the comparison between the numerical and analytical results. The analytical load - displacement solutions are based on the total Green strain and natural strain definitions [CRI 91]. For the solutions based on second PiolaKirchhoff stress and Green strain, the DT and TL-diff results coincide exactly while TL-deri results agree well with the analytical analysis. Results obtained by the UL schemes are shown to correlate with small strain solution. This is not surprising, as it is said in section 2.2., that the UL approaches in this study are restricted to large deformation problems under small strain conditions.
Figure 1. Displacement - force relationships for different
schemes.
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Figure 2. Errors of F for different
465
schemes.
Similarly, DL results agree well with log strain solutions. Overall, the total difference schemes are more accurate than the simple derivation schemes (Figure 2.) because there are less problems of finite increment size used in the numerical procedures, which is assumed to approach zero in the rate equations. Comparisons of CPU times are shown in Figure 3. Times needed for these cases under small deformation in both total strain and incremental approachs are given as a guideline to the extra amount of time needed for large deformation problems.
Figure 3. Comparison of total CPU time required for different
schemes.
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Finite volumes for complex applications
5. Conclusions A numerical method for the analyses of a hypoelastic material is presented. The method is based on the FV discretisation of the law of conservation of linear momentum. Various work conjugate pairs of stress and strain measures are used in constitutive relationships. The developed technique is shown to be simple, easy to understand and manipulate since the underlying concept is simple and straightforward: the balance offerees in the solution domain. This procedure ensures both local and global conservativeness, resulting in meaningful solutions even for coarse numerical meshes. The non-linearity is handled with relatively small additional cost as the method is inherently iterative. This is achieved by using the segregated solution procedure which also offers very efficient memory management. It is clear that the DT approach is the most accurate but it is also limited to materials with single valued stress - strain curves (loading - unloading follow the same path) and is therefore not readily applicable to inelastic problems. The applicability of UL scheme is restricted to small strain cases while the DL approach with Jaumann stress rate is theoretically limited by since the Jaumann rate produces cyclic stresses under shear [KHA 95]. Thus, the TL scheme with 'difference' approach is considered to be the most flexible method; it is accurate and it can be further developed for inelasticity applications. Furthermore, this approach does not require expensive remeshing of the solution domain as the calculation progresses. So far, the method has been successfully employed for simple geometrically non-linear elastic problems. 6.
References
[BON 97] BONET J. et al., Nonlinear Continuum Mechanics for the Finite Element Analysis, Cambridge University Press, 1997. [MCM 75] McMEEKING R.M. et al., Finite-element formulations for problems of large elastic-plastic deformation, International Journal of Solids and Structures, 11, 1975, p. 601-616. [DEM 94] DEMIRDZIC I. et al., Finite volume method for stress analysis in complex domains, International Journal for Numerical Methods in Engineering, 37, N° 2, 1994, p. 3751-3766. [WEL98] WELLER H.G. et al., A tensorial approach to computational continuum mechanics using object oriented techniques, Computers in Physics, 12, 1998, p. 620-631. [CRI 91] CRISFIELD M.A., Non-linear Finite Element Analysis of Solids and Structures volume 1, John Wiley & Sons, 1991. [BAT 96] BATHE K.-J., Finite Element Procedures, Prentice-Hall, 1996. [MAN 99] MANEERATANA K. et al., Finite volume method for geometrically non-linear stress analysis applications, The Seventh Annual Conference of the Association for Computational Mechanics in Engineering ACME'99 (1999), p. 117-120. [KHA 95] KHAN A.S. et al., Theory of Plasticity, John Wiley and Sons, 1995.
A finite volume formulation for fluid-structure interaction. C.J. Greenshields, H.G. Weller and A. Ivankovic
Department of Mechanical Engineering Imperial College of Science, Technology & Medicine London SW7 2BX, UK
ABSTRACT The finite volume method has dominated computational fluid dynamics for many years and has recently emerged as a viable numerical method for stress analysis m solid structures. In this work we have adopted the technique for solving problems involving interaction between fluids and structures. The use of a single code enables simple transfer of information at the fluidstructure interface and an implicit solution procedure for the whole system within each time step. The method is used to study wave propagation in liquid filled, flexible pipes where, since the pressure wave is influenced by the flexibility of the pipe wall, the solution algorithm must be strongly coupled to obtain convergence. Key Words: finite volume; fluid-structure interaction; coupling.
1. Introduction Recent decades have witnessed the development of computational methods to solve problems involving interaction between solid bodies and fluids. Fluidstructure interaction (FSI) covers a wide range of engineering disciplines but a large proportion of all studies fall into the categories of flow-induced vibration, noise and hydrodynamics. Given such a limited scope of application, there is surprisingly little conformity in the numerical methods and coupling procedures adopted throughout research and industry [HAM 95]. The required level of fluid-structure solution coupling depends on the nature of the problem we want to solve. If the deformation of the structure is significant, we would preferably like to solve both components implicitly within each time step. At present, implicit coupling appears hindered by limited availability of a general purpose method capable of treating a variety of problems and by the complexity of
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Finite volumes for complex applications
the methods currently practised. However, the finite volume (FV) method has recently emerged from the realms of computational fluid dynamics (CFD) as a viable alternative for structural analysis [DEM 93;DEM 94]. This presents a new possibility for FSI in which both fluids and structures are solved using the same approach [DEM 95]. The aim of this paper is to illustrate the simplicity and effectiveness of the FV method for FSI modelling through the analysis of wave propagation in liquid filled, flexible pipes. 2. Mathematical Formulation
This work is concerned with a simple mathematical description of the fluid and solid in which energy contributions are ignored. The continuum is governed by the following: • mass balance, or continuity equation
• momentum balance (neglecting body forces)
where p is the density, V is the velocity and cr is the stress tensor. In the case of a linear elastic (Hookean) solid, the continuity equation need not be considered and since deformations are sufficiently small, the convection term in the momentum equation can be ignored and V becomes d\J/dt, where U is displacement. The momentum equation then becomes:
where // and A are Lame's coefficients and I is the identity tensor. For a Newtonian fluid, the continuity equation must be rigidly obeyed and the momentum equation becomes:
where 77 is the fluid dynamic viscosity and p is the pressure. In this work, the system of fluid equations is closed by relating pressure and density for
Fields of application
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compressible flow in liquids by a linearised barotropic relationship
where the subscript '0' represents reference values and K is the bulk modulus. The specification of a problem is completed with the definition of a solution domain and initial and boundary conditions. At the initial instant of time, values of all dependent variables must be specified throughout the solution domain and boundary conditions must be specified at all times either by the value of the dependent variable or the gradient of the dependent variable. 3. Finite volume discretisation
The FV method is based on numerical integration of the system of equations over spatial and temporal domains and can be found in most CFD textbooks [PER 96]. Notable details of the method adopted here are: • face variables in the convection term are calculated using the Gamma differencing scheme [JAS 98] which locally blends second-order accurate central differencing with unconditionally bounded upwind differencing to maintain boundedness; • the PISO algorithm [ISS 96] is adopted to ensure that the velocity field in the momentum equation satisfies the continuity equation. The described discretisation procedure reduces both fluid and solid motion to a set of linear vector equations which are solved in a segregated manner, where each component of the dependent variable is solved separately, treating inter-component coupling terms explicitly. The structure displacements are solved using the Incomplete Cholesky Conjugate Gradient method and the fluid velocity and pressure are solved using the Biconjugate Gradient method [HAG 81]. 4. Fluid-structure coupling
The discretisation method described above has been implemented into the Field Operation and Manipulation C++ library [WEL 98]. The solid and liquid models are combined within a single code to model the transient behaviour of a flexible pipe and a contained compressible liquid. The fluid and structural parts of the solution domain form separate meshes but the interface boundary share
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Finite volumes for complex applications
Figure 1: Implicit solution scheme for fluid-structure interaction the same location in space. The respective systems of equations are solved for both meshes and boundary conditions are passed as shown in Figure 1. 5. Fluid transients in water filled pipes
The test case concerns wave propagation in a water filled unplasticised PVC (PVCu) pipe of external diameter, D = 50 mm and wall thickness, t = 3.8 mm. The dynamic modulus, E — 4.0 GPa and Poissons ratio, v = 0.32 were measured at 20 °C for the PVCu material used in experimental work and its density was 1420 kg/m 3 . The properties of water used in the calculations were K = 2.2 GPa and = 998.2 kg/m 3 . The first simulation looks at the wave speed through water in a straight 200 mm length of pipe. The problem is axi-symmetric and the solution domain is 500 cells in the axial direction by 20 across the fluid and 8 across the wall section; the fluid is given an initial absolute pressure of 2 bar and zero velocity, and is contained at both ends of the pipe. The simulation begins by applying a fluid outlet condition of fixed 1 bar pressure at one end of the pipe. The first 50 //s of wave propagation is presented in figure 2 as pressure along the
Fields of application
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Figure 2: Wave propagation along a flexible pipe pipe axis. At 10 //s the fluid does not see the pipe wall deformation and, as expected, the initial wave travels at the speed of sound in unconfined fluids, c0 = \f~K~J~P — 1485 m/s. By 20 ^zs, the fluid pressure behind the unconfined wave front begins to rise in response to the pipe wall contraction. The initial unconfined wave is quickly suppressed by the surrounding regions of high pressure, as observed in experimentally [THO 69], and a true water hammer wave begins to propagate at an axial speed is 505 m/s. The classical analytical solution for fluid wave speed in an unconstrained, thick-walled pipe [WYL 93] gives a comparable speed of 546 m/s. The analysis only accounts for the change in pipe area which leads to an overprediction in wave speed of 7.5% compared to our numerical solution. 6. Solution stability and convergence
The efficacy of any iterative method is limited by the rate of convergence and worse still, by numerical instability. The coupling procedure adopted here
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Figure 3: Stability of the fluid-structure coupling procedure uses an outer iterative loop over the fluid and structure solver routines whose stability is examined by following the principal steps within a single iterative loop. Initially, the fluid equations are solved on the fluid mesh which has been deformed by the motion of the structure boundary within the previous outer iteration, provoking a change in pressure at the boundary, A.p. The structure equations are then solved and the cells adjacent to the boundary respond to Ap by a displacement Ub- The fluid mesh is then deformed once again and the process is repeated. A condition for stability is that |Ap| must tend to zero with successive iterations, which is illustrated schematically in figure 3. The pressure change at the boundary between successive iterations is depicted as a linear function of the change in displacement for both the fluid and structure. It is clear that if the structure cells are more rigid than the fluid cells at the boundary, the solution stabilises with successive iterations; if the fluid cells are more rigid, the procedure is unstable. In practice, the method is unstable for very flexible structures such as thin-walled plastic containers, hoses and arteries. In such cases, the instability can be eliminated by introducing a form of under-relaxation where, following the structure analysis, the fluid mesh is only deformed by a fraction a of the structural displacement. However, the approach is somewhat contrived since a must be carefully selected for the specific simulation under consideration to ensure reasonable convergence. Rapid convergence is only possible if there is little variation in cell size along the interface and therefore the method is only ideally suited to problems involving simple geometries and small deformations.
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7. Conclusions
Both fluid and structural analyses can be solved in a coupled, fully implicit manner using the FV method within a single code. The method has been validated using the analysis of wave propagation in liquid filled flexible pipes. In our 2 dimensional solution, there are only 5 unknowns and the coupling procedure adopted is easy to understand with the data transfer made simple by the use of one code and the same discretisation procedure for fluid and structure. The numerical prediction of wave speed and peak pressure give good agreement with analytical solutions and experiment and will improve with application of more accurate boundary conditions. Some under-relaxation is necessary to ensure stability of the coupling procedure in systems where the structure is comparatively more compliant than the fluid. The method can be applied to 3 dimensions with minimal additional effort and used to study problems such as water hammer in pipes, blood flow in arteries and impact of fluid filled containers.
Acknowledgements
The authors wish to thank Mr Tom van der Laan for the data from his pipe impact tests. CJG is currently funded by the EPSRC and HGW is funded by Computational Dynamics Ltd. References
[DEM 93]
DEMIRDZIC, I. & MARTINOVIC, D., Finite volume method for thermo-elastic-plastic stress analysis, Computer Methods in Applied Mechanics & Engineering (1993) 109 331-349.
[DEM 94]
DEMIRDZIC, I. fc MUZAFERIJA, S., Finite volume method for stress analysis in complex domains. International Journal for Numerical Methods in Engineering (1994) 37 3751-3766.
[DEM 95]
DEMIRDZIC, I. & MUZAFERIJA, S., Numerical method for coupled fluid flow, heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology. Computer Methods in Applied Mechanics & Engineering (1995) 125 235-255.
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[PER 96]
FERZIGER, J.H.
[HAG 81]
HAGEMAN, L.A. & YOUNG, D.M., Applied iterative methods. New York: Academic Press (1981).
[HAM 95]
HAMDAN, F.H. & DOWLING, P.J., Fluid-structure interaction: application to structures in an acoustic fluid medium, parti: an introduction to numerical treatment, Engineering Computations (1995) 12 749-758.
[ISS 86]
ISSA, R.I., Solution of the implicitly discretised fluid flow equations by operator-splitting, Journal of Computational Physics 62 40-65.
[JAS 98]
JASAK, H. et al, High resolution NVD differencing scheme for arbitrarily unstructured meshes, International Journal for Numerical Methods in Fluids in press.
[THO 69]
THORLEY, A.R.D., Pressure transients in hydraulic pipelines ASME Journal of Basic Engineering (1969) 91 453-461.
[WEL 98]
WELLER, H.G. et a/, A tensorial approach to continuum mechanics using object-oriented techniques, Computers in Physics (1998) 12 620-631.
[WYL 93]
WYLIE, E.B. & STREETER, V.L., Fluid transients in systems, Englewood Cliffs, New Jersey: Prentice Hall (1993).
BOUNDARY CONDITIONS FOR SUSPENDED SEDIMENT Vittorio Bovolin, Luca Taglialatela University of Salerno Department of Civil Engineering Via Ponte Don Melillo 184084 FISCIANO (SA) ITALY Ph. +39 089964087 Fax +39 089964045 Email bovo @ bridge, diima. unisa. it
ABSTRACT This paper deals with the problem of setting appropriate boundary condition for sediment concentration at the channel bed. Sediment transport is a common and very complex phenomenon, in many real situations suspended sediment represents the most important component of the total sediment load. In order to predict sediment load and morphological evolution numerical models are a very useful tool, application of these models requires the setting of boundary condition for the sediment transport equation. In this paper the boundary condition for sediment concentration is set according to a semiempirical relation that links the sediment flux to the local shear stress value. Calculations are carried out using a numerical model solving the averaged Navier-Stokes equations, the standard k-e model is used as turbulence model. Numerical results are compared with experimental data obtained in strong non uniform transport conditions such as: sediment pick-up by a flow initially free of sediment and sediment distribution in a steep sided trench. Key Words: sediment transport, turbulence, boundary condition
^^^^^
1. Introduction Sediment transport by turbulent flows is commonly encountered in practical applications such as rivers, navigation canals, estuaries and reservoirs. Sediments can be transported both as bed load and suspended load, the predominant mode of transport depends on the size, shape and density of the particles in respect to the velocity and turbulent field of the water body. The prediction of sediment transport with the aid of laboratory experiments may be very time-consuming and costly and, for many real life problems, impossible, hence there is a great need for powerful mathematical models to predict it. With the advance of computing methods in fluids,
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it is possible to obtain, at a reasonable computing time, numerical solutions for complex geometry. The boundary condition criteria for sediment concentration at the first grid node of the computational domain close to the bed are not as well established as are for the hydrodynamic counterparts. This paper tests a semi empirical condition proposed by [NAO 97], comparison of numerical result with experimental data shows good agreement.
2. Suspended sediment equation Sediments are subjected to two contradicting actions: velocity fluctuations lift up and keep them in suspension, gravity induce them to settle down. The upward flux of sediment due to the turbulent motion is vc, while the downward flux due to gravity is wsC, where v and c are the fluctuating components, respectively, of the velocity in the transverse direction and concentration, w, is the fall velocity and C is the local volumetric sediment concentration. Assuming an uniform sediment size and a single particle fall velocity ws. the differential transport equation for suspended sediment for steady 2-D turbulent open channel flow is given by:
where V and U are the mean velocity components in the orthogonal and in the flow direction. We consider only cases where the sediment concentration is so low that it does not interfere with the hydrodynamic. In these cases there is no coupling between flow and sediment equations and therefore equation [1] can be solved separately once the mean hydrodynamic field and the settling velocity are known, and when a model for the turbulent sediment diffusion vc has been adopted. In the present paper distribution of velocity, eddy diffusivity as well as friction velocity are calculate by numerical integration of the Reynolds equations with the standard k-e model as closure model. The computational procedure is based on the finite volume method using the SIMPLER algorithm for the pressure coupling [PAT 72], [LAU 74], [PAT 80], [ROD 80] and [VER 95]. Solution of the hydrodynamic model is used as a "frozen" in equation [1]. 3. Boundary condition In computational fluid dynamics it is customary to bridge the layer close to the wall by employing empirical formulae, called wall functions, these formulae provide near-wall boundary conditions for the mean-flow and turbulence transport equations [ROD 80], [VER 95]. These formulae therefore connect the wall conditions to the
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dependent variables at the near-wall grid node. The near-wall grid node is presumed to lie in the fully-turbulent part of the flow. The advantage of this approach is that it escapes the need to extend the computation down to the wall avoiding in this way the need to account for viscous effects in the turbulence model. The wall function approach also includes an option to permit wall roughness effects to be simulated via the specification of an equivalent "sand-grain" roughness height ks [JAY 69]. Once the hydrodynamic is known in order to solve equation [1] it is essential to set the appropriate boundary conditions at the bed. Generally speaking boundary condition can be of the fixed value type (Dirichlet), fixed flux type (Neumann) or mixed one. Fixing sediment concentration or flux at their equilibrium values may be an unrealistic option because, under non equilibrium condition, near-bed concentration and flux change drastically in the downstream direction before the equilibrium value is finally reached. It seems therefore essential to set mixed type boundary condition that may mimic sediment flux variation following changes in entrainment and deposition. Near the bed equation [1] reduces to [CEL 88] and [RAU 90]:
Integrating once yields:
left hand side of equation [3] represents the net flux J of sediment across an horizontal plane. The net flux J is the difference between the entrainment rate E and the deposition rate D. While the latter is equal to wsC the former is not immediately known and therefore it is the main problem in modelling suspended sediment transport in non-equilibrium condition. Starting from equation [3], applying the Reynolds analogy and mixing length concept, [NAO 97] developed a semi empirical boundary condition for uniform and non-uniform situations. Applying Reynolds analogy, [SHI 95] and [ROD 80] yelds:
where lm and /, are, respectively, the mixing lengths of momentum and particle concentration, being the velocity gradient multiplied by the mixing length proportional to the shear velocity, equation [4] can be rewritten as:
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Finite volumes for complex applications
where the proportionality coefficient crc, fixed by the authors, as suggested in literature, equal to 0.50 is known as the Schimdt number, substituting equation [5] in equation [3] a differential equation for C is obtained.
Integration of [6] in the range b
where z = (0cws )/(*"£/*) is the Rouse number and C/ the concentration at the first grid node. To overcome the need to evaluate Cb, it can be assess [NOA 97] that part of the turbulent eddies produced at the upper sublayer penetrates periodically the lower one and picks up sediments at a rate proportional to Q, max v'77, where v' is the root mean square of the vertical velocity fluctuation, replacing v' by £/*, the net flux is expressed as:
where 77 is the picking up efficiency assessed on theoretical basis:
and limited to d+=U*d/v>5. Evaluating Cb from equation [8] and substituting it in equation [7], the net flux J finally becomes:
Equation [10] is the general form for non uniform condition. The expression for uniform condition is obtained setting J=0 in [10], this is equivalent to fix the sediment concentration at the first grid node. Equation [10] can be used once the reference level b, and the concentration Cbjtnajc are specified. To evaluate the bed concentration we follow [RIJ 84] which gave the following expression:
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in which the mobility parameter Tis:
where a' is the standard deviation of effective bed-shear stress, TbiCr,i and Tb,crj are the instantaneous critical bed shear stress, respectively, along and counter local flow direction, J2 and J2 are integrals representing the pick up action of the bed shear stress. The non dimensional grain is:
in which D50 is the particle size, s is specific density (p/p), g the acceleration of gravity and v the kinematic viscosity coefficient (fj/p).
4. Results In order to test the proposed boundary condition we compare numerical results with experimental data obtained by [ASH 82] (as reported by [CEL 84]), by [RIJ 81] and [RIJ 85]. The first two data set ([ASH 82] and [RIJ 81]) refer to experiments in which initially clear water flow, passing over a sand layer, entrains sediment into suspension until the full transport capacity is reached. The second data set ([RIJ 85]) refers to measurements of sediment concentration profiles in steep sided trenches. Table 1 contains all the relevant information for the simulated cases. Case number Reference [RIJ 81] 1 RunT4
h
Um
d
ws (cm/s)))
B/h
Comments
0.230
2.80
0.02
Net entrainment from loose sand bed
k/h
m (mm))
47
0.03
(cm)
(cm/s)
25
2
[ASH 82] RunT5
4.3
37.35
0.15
0.165
1.85
0.15
Net entrainment sand on fixed bed
3
[RIJ 85] T2-T3
39
51
0.06
0.160
1.30
0.01
Net entrainment and deposition in trenches
Table 1. Relevant data for simulated cases
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Finite volumes for complex applications
For sake of simplicity following [RIJ 84] the reference level b has been set equal to O.Old where d is the flow depth. Inlet boundary (x=ff) are of fully developed channel flow. Comparison of calculated and measured sediment concentration profiles allows the assessment of the boundary condition capability to simulate localised effects, furthermore from a practical point of view it is also interesting to verify the capability of the model as a whole to reproduce the total suspended sediment discharge. Figure 1 depicts comparison of calculated sediment concentration profiles with experimental data [RIJ 81], [ASH 82], the general agreement is fairly good. Figure 2a, for the previous data, shows comparison of bottom sediment concentration, it gives an idea of the boundary condition capability to reproduce the pick up action by the flow, particularly noteworthy is the agreement between numerical results and experimental data in the first three sections where stronger is the flow non uniformity. In order to assess the model capability figure 2b reports comparison of calculated and measured total suspended load, also in this case the agreement is certainly encouraging on the model effectiveness to reproduce real life cases. The capability of the model to reproduce sediment concentration adjustment in case of non constant depth is another interesting test on the way to simulate morphological evolution. Figure 3 and 4 depict sediment concentration profiles and total suspended load for a steep sided trenches. This case is particularly demanding because the model has to simulate, in a short stretch, both net deposition, in the enlargement zone, where the flow decelerates and net entrainment in the zone where the flow is under a strong acceleration.
5. Conclusions This paper reports preliminary results of a study on boundary condition for sediment concentration. Numerical results shown fairly good agreement with experimental data. Further research is needed in order to assess the capability of the model to reproduce more complex situation as such as three dimensional case and morphological changes. 6. Acknowledgements The financial support for the research presented in this paper has been provided by a grant of the Italian Ministry for University and Scientific and Technological Research (M.U.R.S.T.) in the framework of the 1997 National Research Project (P.R.I.N.) "Swirling, turbulent and chaotic processes - Water works and Environmental applications".
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7. Bibliography [CEL 88] Celik I. and Rodi W., Modelling suspended sediment transport in non equilibrium situations, Journal of Hydraulic Engineering ASCE Vol. 114, No. 10 (1988) [JAY 69] Jayatilleke C. L. V., Tlie influence of Prandtl number and surface roughness on the resistance of the laminar sublayer to momentum and heath transfer. Prog. Heat Mass Transfer, Vol. 1, p. 193 (1969) [LAU 74] Launder B.E. and Spalding D.B., TJie numerical computation of turbulent flow, Comp. Meth. in Appl. Mech. & Eng., Vol.3, p269, (1974). [NAO 97] Naot D. and Nezu I., Wall functions for the calculation of turbulent 3D sediment transport in open channels XXVH IAHR Congress Vol. 2 San Francisco USA pp. 1268-1273(1997) [PAT 72] Patankar S. V. and Spalding D. B., A calculation procedure for heat, mass and momentum transfer in three dimensional parabolic flows, Int. J. Heat Mass Transfer Vol. 15 pp. 1787 (1972) [PAT 80] Patankar S. V., Numerical heat transfer and fluid flow, Hemisphere publising corporation, Taylor & Francis Group, New York (1980) [RAU 90] Raudkivi A.J., Loose Boundary Hydraulics, Pergamon Press, 3th edition (1990) [ROD 80] Rodi W., Turbulence models and their application in hydraulics, state-of-the-art paper. International Assoc. for Hydr. Res., Delft, The Netherlands (1980) [SHI 95] Shiono K., Falconer R. A., Berlamont J., Elzier M. and Karelse M., A note on statified flow in compound channel, Hydra 2000 XXVIth IAHR Congress, 3B8 pp. 134-139(1995) [RJJ 84] Van Rijn L.C., Sediment transport, part II: Suspended load transport Journal of Hydraulic Engineering ASCE, 110 (11), pp. 1613-1641 (1984) [RIJ 86] Van Rijn L.C., Mathematical modelling of suspended sediment in non-uniform flows Journal of Hydraulic Engineering ASCE, 112, pp. 433-455 (1986) [VER 95] Versteeg H. K. and Malalsekera W., An introduction to Computational Fluid Dynamics,the finite volume method, Longman Scientific & Technical (1995)
Figure L Sediment concentration profiles a) Ashida [ASH 82] b) van Rijn [RIJ 81]
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Figure 2. a) sediment concentration at the bottom b) suspended load [ASH 82]
Figure 3. Sediment concentration profiles in trenches, test2 andtestS [RIJ 85]
Figure 4. Suspended sediment load, respectively, in test2 and test3 [RIJ 85]
Second order corrections to the finite volume upwind scheme for the 2D Maxwell equations
B. Bidegaray
J.-M. Ghidaglia
MIP - CNRS UMR 5640 Universite Paul Sabatier 118 route de Narbonne 31062 Toulouse Cedex France
CM LA - CNRS UMR 8635 ENS de Cachan 61 avenue du president Wilson 94235 Cachan Cedex France
ABSTRACT When computing solutions to Maxwell equations with finite volumes methods one often faces mesh dependent structures. We describe a way to add second order corrections to the finite volume upwind scheme for the 2D Maxwell equations that are designed to overcome this difficulty. This is done by using exact solutions to the wave equation for each component of the electromagnetic field. We illustrate the method with numerical results on simple test cases.
Key Words : 2D Maxwell equations, finite volumes, upwind scheme.
1. Introduction
When computing solutions to Maxwell equations with finite volumes methods one often faces mesh dependent structures. Indeed, fluxes are computed across edges that may have privileged directions in some parts of the computational domain. Our goal is to write a modification of the classical upwind finite volume method that handles this drawback.
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Here we more precisely address the 2D Maxwell equations in the TEZ polarization. Setting Q = (Qi,Q-2,Q3) — (Ex,Ey,Hz} they read
In an homogeneous domain (i.e. e and [L constant) each component of Q satisfies the wave equation
where c2£/z = 1. Both formulations are used to derive our scheme. 2. Leading order terms
The computational domain is decomposed into triangles. Integrating (2) on one of them, K, and on the time interval [0,t], one gets
The computation of IK
is performed through the usual up-
wind scheme (see e.g. [GHI 96]) for hyperbolic systems using formulation (1). More precisely equation (1) is written as Qt + div(AQ) = 0 and IK reads IK = — I JdK
A(n)Qdcr where n is the external unit normal vector on the boun-
dary of the triangle K, dK. The eigenvalues of A(n) are 0 and ±c. For the computation of IK this integral may be split in three contributions considering separately each edge of K. Then we decompose A(n)Q over the eigenvector basis of A(n) and associate Q = Q(K) = to eigenvalue c and Q = Q(-K") to eigenvalue —c where K is K's neighbor across the considered edge of K. This is the usual upwind scheme. This is a particular case of our scheme and we will use it to evaluate its performances.
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3. Second order corrections
The second part,
which happens to be
a second order correction, is computed using the exact solution of the wave equation (2) with initial condition 1 on triangle K and 0 elsewhere. This approach has already been used by Abgrall [ABG 94], Gilquin et al [GIL 91, GIL 94] and Cha'ira [CHA 95] for solving the Riemann problem for gas dynamics and writing Euler equations as a wave equation, but in our case we have no conformal invariance. insertion of UQ which is formally bound to disappear in the next differentiation ensures that we preserve a constant solution over space (and time) if the initial data is constant, which is the first step towards flux conservation. We notice that this function w is solution to the wave equation
with initial data WQ = 0, initial time derivative w\ = UQ — UQ(K] and right hand An exact solution to this wave equation is given by Kirchoff formulae
and H denotes the Heaviside function. Hence the computation of UK leads to multiple integrals on the edges of the triangles that read (C and C1 being equal or adjacent edges)
and
that may be expressed after tedious calculations by means of classical functions of the variables (lengths and angles in the triangle). We make an approximation here by computing only integrals over edges of K or its neighbors but not
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Finite volumes for complex applications
triangles that only have one vertex in common with K, otherwise the computation of UK would be exact. This part of the derivation of our scheme is specific to the 2D case since the fundamental solution G to the wave equation has different expressions in different dimensions. The ID case is easy to compute but is of no interest since there is no point in correcting direction dependent structures. After this modification the scheme is no longer a finite volume scheme : there is no exact balance of fluxes. The formula for UK shows some discontinuity at time and if we denote by a the angle between edges C and C when are distinct, a acute
a obtuse
W0
W These formulae are continuous for a — | and are a puzzle for computer algebra systems. To obtain explicit formulae and avoid numerical integration is a major advantage for numerical computations. This computation is performed with a condition on the size of triangles and the time step (that implies, in particular that no length appears in the final result). More general conditions may be taken into account but then the integration should be performed on more triangle edges. To take account of the discontinuity at time t = 0, we do not use the full correction but only a fraction of it given by a factor 0 6 [0,1]. This means that we make a balance between time t and time 0 where UK — 0. Since we deal with a second order correction this does not affect the consistence of the scheme, whatever the choice of 9 is. The CFL ratio and this parameter 0 are the two parameters to tune in order to obtain a "good" scheme. They are strongly linked in ID but may be chosen independently in 2D. Numerical results show that 9 = 0 is not the best choice with respect to norm conservations for example. But the best 9 also depends on the mesh and is therefore difficult to choose.
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4. Numerical results This work is still in progress and up to now only simple test cases have been performed. Different boundary conditions have been implemented (following [ENG 77, ENG 79, JOL 89], namely absorbing boundary conditions, perfect conducting surfaces and perfect reflecting boundaries. Different initial conditions and incident fields have also been tested. The main drawback of the second order correction is that it induces some extra dissipation. This is very easy to see on a ID equivalent of our correction. This dissipation is larger for a large 9. This leads us to choose 0 E [0, .3] for numerical simulations. The best conservation of norms is usually obtained in this interval. The following figure shows that for a test where the L2 and L00 norm is supposed to be conserved, the choice the upwind scheme without correction (9 = 0) is not the better scheme for norm conservation. On Fig. 1 only the L°° norm of Ex is represented. The different curves correspond to different meshes for the same test case. Curves are similar for other components of the field or the L2 norm. The fact that curves are decreasing is not generic but specific to this field in the computed case. Our correction seem to benefit finer meshes but criteria to adjust 9 to a particular mesh are not obvious to find. Since we only add a small perturbation to the original scheme we can not expect any real improvement of its main defaults, like phase shift. The initial purpose of the introduction of our UK was the correction of mesh dependent structures and we have tested the propagation of a wave front with different angle of incidence. For small angles (otherwise our boundary conditions have to be improved) we show a real improvement of the straightness of the front. This shown on Fig. 2 and 3.
5. Perspectives Perspectives of this work may be found in different directions. First we may go towards more realistic test cases and try to model real physical structures. A complete study of the case of an inhomogeneous media has to be performed. This would also contribute to more physical test. The 3D case is also of interest and formal calculations of the correction have to be derived in this context.
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Finite volumes for complex applicationsions
6. Bibliography [ABG 94]
[CHA 95]
[CIO 93]
[ENG 77]
[ENG 79]
[GHI 96]
[GIL 91] [GIL 94]
[JOL 89]
ABGRALL R., «Approximation du probleme de Riemann vraiment multidimensionnel des equations d'Euler par une methode de type Roe (I) and (II)», C.R. Acad. Sci. Paris 319, 1994, p. 499-504 and 625-629. CHAIRA S., Sur la resolution numerique des equations d'Euler de la dynamique des gaz par des schemas multidimensionnels, PhD Thesis, Ecole Normale Superieure de Cachan, 1995. CIONI J.-P., FEZOUI L. AND STEVE H., «A parallel time-domain Maxwell solver using upwind schemes and triangular meshes», Rapport INRIA 1867, 1993. ENGQUIST B. AND MAJDA A., «Absorbing Boundary Conditions for the Numerical Simulation of Waves», Math. Comput, 31, 1977, p. 629-651. ENGQUIST B. AND MAJDA A., «Radiation Boundary Conditions for Acoustic and Elastic Wave Calculations», Commun. Pure Appl. Math., 32, 1979, p. 313-257. GHIDAGLIA J.-M., KUMBARO A. AND LE COQ G., «Une methode Volumes finis' a flux caracteristiques pour la resolution numerique des systemes hyperboliques de lois de conservation», C. R. Acad. Sci., seme I, 332, 1996, p. 981-988. GILQUIN H. AND LAURENS J., «Problemes de Riemann multidimensionnels pour les systemes hyperboliques lineaires», 1991. GILQUIN H., LAURENS J. AND ROSIER C., «The explicit solution of the bidimensional Riemann problem for the linearized gaz dynamics equations», ENS de Lyon, UMPA 133, 1994. JOLY P. AND MERCIER B., «Une nouvelle condition transparente d'ordre 2 pour les equations de Maxwell en dimension 3», Rapport INRIA, 1047, 1989.
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FIG. 1: Norm conservation The L°° norm is represented as a function of 9 and for different meshes, the exact L2 norm being the horizontal line. The 'cross'-curve correspond to the finer irregular mesh (others are 'star' and 'circle'-curves) and the 'diamond'curve is a regular mesh that is finer than the 'square'-curve.
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Finite volumes for complex applications
FIG. 3: Front straightness : 0 — 0.1
A MHD—Simulation in Solar Physics A. Dedner^, C. Rohde^, M. Wesenberg^ t Institut fur Angewandte Mathematik, Universitdt Freiburg, Germany * Centre de Mathematiques Appliquees, Ecole Polytechnique, Palaiseau, France
ABSTRACT A Riemann-solver-based FV-scheme for the numerical solution of the equations of magnetohydrodynamics(MHD) in two and three space dimensions is proposed. Using recently developed techniques for the efficient combination of high-order resolution, adaptive mesh refinement and parallel computing the method is designed for complex flow problems in solar physics. Here we consider the rise of patterns of concentrated magnetic fields in the sun's convection zone. While rising to the solar surface these so-called flux tubes are fragmented by instabilities of Kelvin-Helmholtz- and Ray leigh-Taylor-type. Key Words: MHD, Riemann solver, high-order FV-schemes, adaptive mesh refinement, magnetohydrodynamical instabilities.
1. Introduction In the last decade the numerical solution of hyperbolic and hyperbolicparabolic conservation laws made considerable progress. In particular this applies to the case of several space dimensions. Features like upwind techniques, shock capturing, and formally high order resolution are standard now, even on unstructured grids. Furthermore the concepts of adaptive mesh refinement using grid indicators respectively error estimators and grid alignment became widely used. On the level of model problems a lot of corresponding analytic results on convergence and rigorous error estimates are available. Concerning physically relevant problems it is noteworthy that these techniques —despite their general applicability— have been utilized merely for the system of gas dynamics. Up to now, less has been done for more complex situations. Motivated by problems of solar physics this paper adresses two issues. First we present a numerical scheme that extends the methods mentioned above to the equations of ideal magnetohydrodynamics (MHD) in Section 2. Specific difficulties caused by the structure of the MHD-system are discussed, for in-
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Finite volumes for complex applications
stance the divergence free condition for the magnetic field. The second issue of this paper is the presentation of numerical results for phys ically relevant two and threedimensional MHD-simulations. We start with th discussion of two basic problems: the Kelvin-Helmholtz instability and th Rayleigh-Taylor instability (Sections 3.1 and 3.2). Finally we present result of ongoing numerical studies for a problem in solar physics which is strong! influenced by these instabilities (Section 3.3). To be specific we study th question how far a so-called flux tube, that is a pattern of concentrated mag netic field, can rise to the solar surface. If a flux tube can rise far enough thi would support the conjecture that flux tubes cause the sun spots which can b observed from earth on the solar surface. 2. Numerical Solution of the MHD-System in 2D/3D 2.1. The MHD-System The MHD-system describes the interaction of the motion of a compressible electrically conducting fluid with a magnetic field. For a spatial domain Q C R3 it is given by[Ca65, Ch61]:
Here p ( x , t ) stands for density, u(x,t) for velocity with components ux,uy,uz, B(x, t) for the magnetic field with components Bx,By,Bz, and E for the total energy. For the pressure p ( x , t ] , the matrix P(x,t) — ( p i j ( x , t ) ) e R 3 x 3 is defined as
To close the system, we add the thermodynamical relations
where T(x,t} is the temperature and e = £(p,T) the internal energy. Note that, for g(x,t) e R, the effects of gravity are included in the system above. We recall that the MHD-system above can be written as a system of hyperbolic
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conservation laws with the vector of unknowns U := (p,pu,B,pE) 6 U C R8, fluxes F, G, H : u ->• R8 and source 5 : ft x R>0 x U -> R8:
Initial and boundary conditions will be specified when needed. 2.2. The Numerical Scheme
In this section we will present our scheme to solve the MHD-equations numerically. For sake of simplicity we will restrict the presentation to a polygonal domain 0 C M2 that is covered by a grid of triangles. A part of the grid is displayed in Figure 1 to introduce notations.
Figure 1. Triangular grid and Notations. For initial data Uo : 17 —>• U, the FV-scheme takes the form
Here Atn denotes the local time step constrained by a suitable CFL-condition and \Tj\ the area of Tj. As usual, the numerical flux gjl is thought to be an approximation of
where ni = ( n j l , . . . , n*-jl) E R8, i = 1, 2. The term results from integrating the conservation law with respect to the time interval and the triangle Tj. Using the rotational invariance of the MHD-system the integrand can be expressed solely in terms of the flux F:
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Finite volumes for complex applications
This reduction to a onedimensional setting allows us to use (approximate) Riemann solvers. Based on an extensive study of different solvers in the MHDcase we favor the one suggested by Dai and Woodward, since it offers a good compromise between computational effort and accuracy [We98]. For details on this scheme we refer to [DW95]. By the linear reconstruction technique of Durlowsky, Engquist and Osher and the use of a second order Runge-Kutta method the basic FV-scheme described above has been extended (formally) to second order. To allow local refinement and coarsening of the grid we have to introduce an error estimator or a grid indicator. Error estimators for nonlinear hyperbolic systems in multiple space dimensions seem to be out of reach for the moment. For the Euler equations density-based indicators are well-established. Note that, in addition to the density, the magnetic field has to be taken into account for MHD. Even in one space dimension there exist discontinuous solutions of the MHD-system (Alven waves, entropy wave) which do not undergo a jump in density and the magnetic field. Extending a grid-indicator for the Eulerequations we propose the following grid indicator for refinement respectively coarsening.
The triangle Tj has to be refined, if ref_ind(Tj) > ref_limit, and may be coarsened, if crs_ind(Tj) < crs_limit, where ref_limit, crsJimit > 0 are some threshold values. Another MHD-specific problem is caused by the condition that V-B has to vanish. While this property is conserved by exact solutions for initially divergence free magnetic fields, it is not automatically satisfied in numerical schemes. A disregard of this problem can lead to severe numerical problems [BB80]. As a remedy we add an artificial source term, which is proportional to V • B [As96, Po94]. In our experience this approach works well in many situations but seems to have merely a stabilizing effect and does not necessarily reduce the divergence error. In particular problems with stagnation points remain a challenge. Even the computational advantages, that we achieve by local adaptation and higher order schemes, do not suffice to resolve small scale structures using a one-processor machine. Therefore our 2D-Code has been implemented on a parallel computer with a shared memory architecture. This gave good speedup results for large grids. Unfortunately computers with shared memory architecture and a large number of processors are not easily available. In 3D we use a code which can be run on machines with distributed memory by using MPI to communicate. This code is based on a grid concept which was developed by
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Schupp [Sc99]. To gain still from local grid refinement a concept for dynamic load balancing is integrated. 3. Numerical Results
As outlined in the introduction of this paper our code is designed for the simulation of problems in solar physics. In particular we are interested in the rise of flux tubes in the sun's atmosphere. These strong magnetic fields are assumed to develop in the deeper convection zone. When they rise they are subject to instabilities which break them up and slow down their ascend. One type of instability is caused by the lower density in the interior of the tube compared to the density of the atmosphere through which it rises. This results in Rayleigh-Taylor instabilities which increase the boundary of the tube and decrease its rise to the solar surface. At the same time the sides of the tube are fragmented by Kelvin-Helmholtz instabilities due to the shear flow between the atmosphere and the rising tube.
Figure 2. The time evolution of a KH instability for t — 0.4,0.6,0.8,1.0 using 256218,475374, 571044, 727138 elements respectively. Starting with [Ch61] both types of instabilities have been studied thoroughly in the literature for their own sake. The results show that the amplification of small scale disturbances can be reduced or even suppressed by a magnetic field tangential to the separating interface of the fluids. The magnetic field can be
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Finite volumes for complex applications
used to increase the stability of the rising flux tube, which leads to a rise into higher regions of the atmosphere without being too strongly fragmented. This stabilization is obtained by twisting the magnetic field in the flux tube so that it is tangential at the boundary. For physical background we refer to [Sch79]. In our calculations we show the performance of the code for pure KelvinHelmholtz- and Rayleigh-Taylor instabilities. Then we present two results for the flux-tube problem: with and without stabilization by twisting. For a quantitative study consult [DRW99]. 3.2. Kelvin-Helmholtz Instability (KH) In the first problem we consider a KH instability in a rectangular domai n that is separated in two parts by a horizontal line. For the initial conditior in fJ, we take The velocity in x-direction is ux0 = ±5 in the upper/lower part of n. To start the instability we perturb uy periodically. As boundary conditions we choose periodic conditions for the vertical boundaries and outflow conditions for the horizontal boundaries. The results for the transport of the density with the calculated velocity field at different time levels are displayed in Figure 2. For visualization we use a special tool to treat vector fields[BR98]. Note that gravitation is not included.
Figure 3. RT instability at time t = 4.5 for Bx/ f4 = 0.00,0.01,0.05,0.1.
3.3. Rayleigh-Taylor Instability (RT) The second test case shows a perturbed horizontal layer of lower density in a stratified atmosphere. Under the influence of gravitation this layer rises. Because of a perturbation of the initial values it produces the typical " finger"structure of the RT-type instability. We show the announced stabilizing influence of a growing tangential magnetic field in Figure 3 (Displayed quantity:
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density). As boundary conditions we have periodic conditions for the vertical conditions. For the the horizontal boundaries and the construction of the atmosphere we refer to [DRW99]. 3.4. Rise of flux tubes to the solar surface We turn to calculations of a flux tube in 2D. In the top row of Figure 4 we show By for a flux tube with a magnetic concentration normal to the plane of calculation. This magnetic field cannot reduce the fragmentation through the instabilities so that the flux tube is strongly fragmented and hardly rises.
Figure 4. Flux tube in 2D. In the lower row of Figure 4 the calculation is repeated with a twisted magnetic field. This prevents the fragmentation and allows the flux tube to rise faster and higher than in the first calculation. Figure 5 shows a first calculation for a flux tube in three space dimensions.
Figure 5. Flux-tube in 3D for a fixed time t — 0.6. Top: i/z-plane for x = 0, bottom: xz-plane for y = 1, y = 3.3, y = 5.6.
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ACKNOWLEDGEMENTS The authors were partially supported by the DFGSchwerpunktprogramm "Analysis und Numerik von Erhaltungsgleichungen" and the EU-TMR research network for Hyperbolic Conservation Laws. Bibliography [As96]
ASLAN N., Two-Dimensional Solutions of MHD Equations with an Adapted Roe Method, J.Num.Meth.Fluids 23, 1996, p. 1211-1222.
[BR98]
Becker J., Rumpf M.,, Visualization of Timedependent Vector Fields by Texture Transport Methods, In Proceedings of the Eurographics Scientific Visualization Workshop '98, 1998.
[BB80]
BRACKBILL J.U., BARNES D.C. The effect of nonzero V • B on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys. 35, 1980, p. 426-430.
[Ca65]
CABANNES H., Magnetodynamique des fluides, Paris (1965).
[Ch61]
CHANDRASEKHAR S., Hydrodynamic and hydromagnetic stability, Oxford (1961).
[DW95]
DAI W., WOODWARD P.R., A Simple Riemann Solver and High-Order Godunov Schemes for Hyperbolic Systems of Conservation Laws, J.Comp.Phys. 121 (1), 1995, p. 51-65.
[DRW99]
DEDNER A., ROHDE C., WESENBERG M., A numerical study on magnetohydrodynamic instabilities in three space dimensions, in preparation.
[Po94]
POWELL K.G., An Approximate Riemann Solver for Magnetohydrodynamics (That Works in More than One Dimension), ICASE-Report 94-24.
[Sc99]
SCHUPP B., Entwicklung eines effizienten Verfahrens zur Simulation kompressibler Stromungen in 3D auf Parallelrechnern, PhD-thesis, Freiburg (1999).
[Sch79]
SCHUSSLER M., Magnetic Buoyancy Revisited - Analytical and Numerical Results for Rising Flux Tubes, Astronomy and Astrophysics 71, No. 1-2, 1979, p.79-91.
[We98]
WESENBERG, M., A Note on MHD-Riemann-Solvers, print (1999).
Pre-
A Zooming Technique for Wind Transport of Air Pollution* P.J.F. Berkvens, M.A. Botchev, W.M. Lioen, J.G. Verwer CWI, P.O. Box 94079 1090 GB Amsterdam, The Netherlands http://www.cwi.nl/ e-mail: [berkvens, botchev, waiter, janv]@cwi.nl ABSTRACT In air pollution dispersion models, typically systems of millions of equations that describe wind transport, chemistry and vertical mixing have to be integrated in time. To have more accurate results over specific fixed areas of interest—usually highly polluted areas with intensive emissions—a local grid refinement or zoom is often required. For the wind transport part of the models, i.e. for finite volume discretizations of the transport equation, we propose a zoom technique that is positive, mass-conservative and allows to use smaller time steps as enforced by the CFL restriction in the zoom regions only. KEY WORDS: finite volumes, advection schemes, local refinement, air pollution, high performance computations 1
Introduction
Mathematical problems often encountered in air pollution modelling are transportreaction problems of the form
where cs are the concentrations of m chemical species in the atmosphere. The species under consideration are not only pollutants, but all the main chemical substances present in the atmosphere. In real applications m lies between 25 and 100. The first term in (1) denotes the time rate of change of cs, the term V • (ucs) describes the transport of the species by a given wind field u. The term V(u, cs) on the right-hand side appears as a result of parameterization of transport processes not resolved on the *This work has been done within the program "Wiskunde Toegepast" ("Mathematics Applied") of NWO, the Netherlands Organization for Scientific Research, project no. 613-302-040. This work was supported by NCF, the National Computing Facilities Foundation, under Grant NRG 98.02.
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grid. It will be referred to as vertical mixing. Vertical mixing is often modelled by means of turbulent diffusion parameterization. The stiff reaction term R describes the chemical reactions among the species cs. Usually, (1) is discretized in space with a finite-volume technique. For typical grid resolutions and numbers of chemical species ra, (1) yields a system of millions of equations that has to be integrated in time. More background information on air pollution modelling can be found in the recent books [4, 11] and the survey [7] giving an overview of numerical techniques used in the field. 1.1 Model In air pollution models problem (1) is typically discretized in space with a finitevolume technique and then integrated in time with operator splitting. This means that the whole advance in time consists of separate advection, vertical mixing and chemistry advances. Chemistry and vertical mixing are integrated in time implicitly, to avoid a severe restriction on the time step At imposed by the chemistry-mixing stiffness. Advection is usually integrated in time explicitly. In a particular atmospheric dispersion model we are working with, a successor of [5], several modern advection schemes are used. The two basic schemes are the Slopes scheme [9] and the Split scheme, a third order flux-limited upwind scheme [6, 8]. Both schemes are onedimensional and applied with directional splitting. For the atmospheric application it is important that mass is conserved and concentrations remain positive (cs(t) ^ 0, t ^ 0). Both advection schemes mentioned have these properties on uniform grids, even for divergent flows, which occur in the atmosphere. Positivity of species concentrations is guaranteed for time-step size restrictions such that no negative air masses can occur during any one of the split advection steps, which is a natural constraint. 1.2 Why zooming? To capture local phenomena—as occurring over highly polluted areas—without increasing the cost too much, it is often desirable to have a nonuniform grid with local refinement (zooming) over the areas of interest. The local grid refinement or zoom methods have been subject of active research (see e.g. relatively early works [1, 3]). Most of the effort has been to develop powerful adaptive grid refinement. In our case, however, adaptivity is not needed and not even welcome. First of all, areas of interest are known in advance (that can be Europe, for example). Second, meteorological data as e.g. wind fields are often not available on a fine grid in the whole domain (see Figure 1). Third, the position of the zoom regions is often determined by the chemistry part of the model: for example, one has to refine in the regions with high emission activity. Thus, our goal is an efficient robust zoom algorithm for fixed zoom regions and we propose a simple strategy for this. 2 Zooming technique 2.1 Requirements for zooming Since mass conservation and positivity are important for the atmospheric application, we wish our zooming technique to preserve both these properties (something we have
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Figure 1: An example of zooming. Zoom regions are numbered. not managed to find in the literature). Due to the smaller size of the finite volumes, the CFL restriction is more severe in zoom regions and a smaller time step At needs to be taken. Positivity and mass conservation are more difficult to preserve if we want to have smaller step sizes in the zoom regions only, so that e.g. in a zoom region with refinement factor two, two time steps are to be done within each global advance in time. Our zooming technique gives a simple and reliable way to have smaller timestep sizes in the zoom regions only while preserving positivity and mass conservation across zoom interfaces. 2.2 How to zoom in ID We explain first how to perform zooming for the one-dimensional (ID) transport equation ct + (uc)x = 0 and then how to extend this approach to the three-dimensional (3D) case. We use advection schemes in the mass conservative flux formulation. Each cell i contains a mass fa being the integral of c over the zth cell volume. In 1D fluxes Fi+1/2 are calculated giving the amount of chemical species transported per time step between cells i and i + 1. They depend on the given air-mass fluxes determined from the velocity field and on the masses fa. The time advance has the form
where Fi ±1/2 are the fluxes at time level n. On the boundaries of the zoom region its advection scheme is adjusted to the scheme on the coarse level. As an example, consider the situation with refinement factor ref = 3 (see Figure 2). This adjustment consists of two elements. First, the three outermost cells of the zoom region are lumped together and considered to be one single cell (the interface cell). The tracer mass in the interface cell is assumed to be distributed uniformly (or in accordance with its slope for the slope scheme) over its fine grid subcells, and the fluxes between these subcells are never computed. Second, on the boundary of the zoom region we calculate the flux Fc that is coarse in both space and time. For each coarse time step, there are three corresponding time substeps that have to be done in the zoom region. The coarse boundary flux Fc is applied at
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Figure 2: 1D zooming: one time step tn —» tn+1 corresponds to three time substeps in the zoom region. The "coarse" flux Fc is applied on the first substep. once at the first fine time substep, for the second and for the third substep zero flux is taken at the boundary of the zoom region. It can be shown that this simple strategy guarantees mass conservation and positivity of the chemical species for many modern advection schemes including the Slopes and Split schemes (for more details see [2]). The time-step restriction for positivity is basically the same as in the uniform case, namely that the coarse air mass flux on the one wall plus the three fine air mass fluxes on the other wall do not take out more mass than there was in the cell. The strategy is also easily generalized to the multidimensional case, with a similar time-step restriction per direction. Another choice would be to equidivide Fc in three parts and apply them consecutively at the same times as the fine fluxes on the other side of the interface cell. In ID as well as in multidimensional cases this gives no longer a positive scheme, unless a more severe time-step restriction is enforced. The problem is that parts of the coarse flux, which was calculated with the old concentration field, are applied to a new concentration field. The latter may have changed in such a way that positivity is lost. 2.3 How to zoom in 3D We now give more details about our zooming algorithm in 3D. For simplicity reasons we restrict ourselves to block-shaped zoom regions in 3D. For our applications this is sufficiently general. A zoom region has interface cells all along its edges forming the boundaries with its parent region. The ID algorithm described above can then be applied in a split manner. The only complication as compared with the ID case is caused by the fact that the advection scheme is directionally split. We use a Strang (symmetric) splitting scheme consisting of six split steps as follows: X-Y-Z-Z-Y-X, where X denotes the application of the advection operator in the z-direction during half a time step, At/2, etc. For one triplet of advection steps—one in each direction—on the coarse level, e.g. X-Y-Z, in the zoom region we perform ref triplets of advection steps in the same and the opposite order, thus x-y-z-z-y-x in case ref = 2. Here x denotes the application of the advection operator in the re-direction during a quarter time step, At/4, etc. For example, in Figure 3 we show the sequence of the directional substeps in a zoom region with refinement factor 2. Clearly, Strang splitting order is preserved in the zoom region, which is important for accuracy reasons. The coarse fluxes computed by the parent at the boundary with the child region
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Figure 3: The sequence of advection steps in a coarse and a fine region with refinement factor 2. are used as boundary conditions for the child region and are applied at once at the first time substep per direction. After the zoom or child region has carried out its ref triplets, the concentration field in the child region is copied back to its parent after suitable coarsening. This then allows the parent region to advance another three directional substeps and to provide new boundary conditions for the child region. This 3D zooming algorithm can also be shown to be mass-conservative and positive for the advection schemes we are interested in. The complete description of our algorithm can be found in [2]. 3 Experiments With our tests we want to demonstrate that (i) our zooming technique gives an accuracy comparable with the accuracy one would get on the overall fine grid provided that important phenomena remain in the zoom regions; (ii) this accuracy is achieved with significantly less computational effort; (iii) even when important phenomena occur outside the zoom regions, our algorithm still gives a solution which is at least as accurate as one would get on the overall coarse grid. The situation (i) is quite common for atmospheric modelling since one of the typical needs for the local refinement is concentration of the emission sources in highly polluted areas. We note in passing that zooming also leads to a more economical memory use, compared to the overall fine grid. We present 2D numerical results obtained with our zoom implementation for the Slopes advection scheme. As a model problem for our experiments we take the solidbody rotation test commonly known as the Molenkamp-Crowley test [10]. In this test the advection equation on the (unit) sphere
is solved for the velocity field u = 27rcosa:sin?/, v = -27r sin x. Here x E [0; 2yr] and y E (—Tr/2; Tr/2) are the longitude and the latitude coordinates respectively. The chosen velocity field provides a solid-body nature of the air rotation over the sphere, so that after one full rotation (at time t = 1) the solution will coincide with the initial distribution c(x, y, 0). For the initial distribution in our tests we take a cone of height 1 centered at the point (x, y) = (37T/2,0). We stress that the cone provides a severe test for any advection scheme. The radius of the cone base is taken to be 21 grid cells with respect to the finest zoom region used. In Figure 4, we show the velocity field and the position of the zoom regions used in the tests. Overall, we have a nonuniform grid consisting of three regions: the coarsest,
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Figure 4: The velocity field and the position of the zoom regions. with resolution Ax = Ay = 4.5° and two zoom regions refined with factors 2 and 6 (resolutions Ax = Ay = 2.25° and Az = Ay = 0.75° respectively). First of all we have compared the zoom grid solution with the solution obtained on the overall fine grid (with the same resolution as in the finest zoom region). As desired, it turned out that the difference between them is negligible as long as the moving cone remains in the zoom region. The difference in the solutions is explained by the fact that the order of the directional substeps in the zoom algorithm alternates (as in Figure 3) whereas it remains the same for the uniform grid algorithm. Performing one full rotation (0 ^ t ^ 1) on the nonuniform zoom grid means that the cone has to travel over the whole sphere, leaving and then entering again the zoom regions. Of course, with the local zoom regions fixed in space, i.e. not moving with the cone, globally one can not expect much better accuracy than on the overall coarse grid. This is confirmed in our tests. We have performed one full rotation on the overall coarse (Ax; = Ay = 4.5°), the zoom, and the overall fine (Ax = Ay = 0.75°) grids. In each case, we have measured accuracy by comparing the solution cn after one full rotation with the initial distribution c° (the exact solution would give no difference with c°). We have computed the following errors (representing minimum, maximum, scaled /2, mean and variance errors respectively):
where rj is the (i, j)-cell air mass divided by the total mass of the air (cells near the poles have less mass), and the sums and the minima/maxima are taken for all the grid cells within the finest zoom region. To be able to compare errors of different space resolutions, on the overall fine and on the zoom grids we computed the errors on the data coarsened up to the coarse grid resolution. We have summarized our accuracy observations in Table 1. In Figure 5 (left picture), we plot the zoom solution after the full rotation. As we
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Table 1: Errors. "Local error" means with the cone cone still being within the finest zoom region, "global error" means after one full rotation Errors fine zoom coarse
local: ^global same as fine
emin -5.9e-3 -1.8e-2 -1.9e-2
emax -3.1e-2 -0.12 -0.21
global: errO 2.5e-3 1.7e-2 3.3e-2
errl 1.4e-3 1.8e-3 2.6e-3
err2 -3.5e-5 -0.12 -0.15
Figure 5: The cone after the full rotation see, the cone shape is strongly deformed. This is not only because it has travelled through the grid that is 6 times coarser (we should emphasize that the resolution 4.5° is quite coarse for this test) but also because of the accuracy losses near the poles. For comparison reasons, at the right plot we present the solution for the same test but with half the grid size in the coarse region (in other words, cone is to travel through the grid that is only factor 3 coarser, with basic resolution Ax = Ay — 2.25°). As expected, deformation is significantly decreased. Finally, we comment briefly on the computational expenses. Rough estimates taking into account total number of grid cells and the CFL restriction on the time step size show that the computations on the overall fine grid would be 5.2 times more expensive than on the zoom grid. This is however a too optimistic speed-up estimate which does not take into account communication overhead for the zoom algorithm. With the current implementation (SGI workstation) we observed that our zoom algorithm is approximately 2.6 times faster than the uniform algorithm on the fine grid. However, code optimization has to be performed yet. 4 Conclusions We have presented a positive and mass-conservative local grid refinement (zoom) algorithm for advective transport. The algorithm can be applied to many modern advection schemes with directional splitting in space and explicit advance in time. With our approach, a smaller time step (due to the stricter CFL condition) is taken within the zoom regions only.
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The described zoom algorithm has been implemented as a code that allows to use an arbitrary number of zoom regions that can lie inside each other in an arbitrary way, provided they are of block shape and strictly embedded. To have high performance of the uniform grid code preserved as much as possible, the work of the zoom code is organized in a uniform region-wise manner, i.e. the zoom grid is split into a cascade of uniform grid regions. For more details on the implementation we again refer to [2]. Our zoom code will be incorporated in the air pollution model TM3, a recent successor of [5]. The TM3 model is operational at the Institute of Marine and Atmospheric Research (IMAU, Utrecht University), the Dutch Royal Meteorological Center (KNMI) and the Dutch National Institute of Public Health and the Environment (RIVM). References [1] M. J. Berger. Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations. PhD thesis, Department of Computer Science, Stanford University, Stanford, CA 94305, Aug. 1982. [2] P. J. F. Berkvens and M. A. Botchev, W. M. Lioen and J. G. Verwer. A zooming technique for wind transport of air pollution. Report MAS-R99xx, CWI, 1999. [3] J. H. Flahery, P. J. Paslow, M. S. Shephard, and J. D. Vasilakis, editors. Adaptive methods for Partial Differential Equations. SIAM, Philadelphia, PA, 1989. [4] T. E. Graedel and P. J. Crutzen. Atmosphere, Climate and Change. Scientific American Library. Freeman and Company, New York, 1995. [5] M. Heimann. The global atmospheric tracer model TM2. Technical Report 10, Deutches Klimarechenzentrum (DKRZ), Hamburg, 1995. [6] W. Hundsdorfer and B. Koren, M. van Loon and J. G. Verwer. A positive finitedifference advection scheme. Journal of Computational Physics, 117:35-46, 1995. [7] J. G. Verwer, W. Hundsdorfer and J. G. Blom. Numerical time integration for air pollution models. Report MAS-R9825, CWI, 1998. [8] A. C. Petersen, E. J. Spec, H. van Dop, and W. Hundsdorfer. An evaluation and intercomparison of four new advection schemes for use in global chemistry models. Journal of Geophysical Research, 103(D15):19,253-19,269, Aug. 1998. [9] G. L. Russell and J. A. Lerner. A new finite-differencing scheme for the tracer transport equation. J. Appl. Meteor., 20:1483-1498, 1981. [10] D. L. Williamson and P. J. Rasch. Two-dimensional semi-Lagrangian transport with shape-preserving interpolation. Monthly Weather Review, 117:102-129, 1989. [11] Z. Zlatev. Computer treatment of large air pollution models. Kluwer Academic Publishers, 1995.
Computational Solid Mechanics using a Vertex-based Finite Volume Method
G. A. Taylor, C. Bailey and M. Cross Centre for Numerical Modelling and Process Analysis University of Greenwich, Woolwich, London SE18 6PF, UK E-mail: [email protected]
ABSTRACT A number of research groups are now developing and using finite volume (FV) methods for computational solid mechanics (CSM). These methods are proving to be equivalent and in some cases superior to their finite element (FE) counterparts. In this paper we will describe a vertex-based FV method with arbitrarily structured meshes, for modelling the elasto-plastic deformation of solid materials undergoing small strains in complex geometries. Comparisons with traditional FE methods will be given. Key Words: Vertex-based, Finite Volume, Solid Mechanics, Elasto-plastic.
1. Introduction Over the last three decades the FE method has firmly established itself as the pioneering approach for problems in CSM, especially with regard to deformation problems involving non-linear material analysis [OH80, ZT89]. As a contemporary, the FV method has similarly established itself within the field of computational fluid dynamics (CFD) [PatSO, Hir88]. Both classes of methods integrate governing equations over pre-defined control volumes [PatSO, Zie95], which are associated with the elements making up the domain of interest. Additionally, both approaches can be classified as weighted residual methods where they differ with respect to the weighting functions that are adopted [OCZ94]. Over the last decade a number of researchers have applied FV methods to problems in CSM [Tay96]. It is possible to classify these methods into two approaches, cell-centred [DM92, HH95, Whe96, Whe99] and vertex-based [FBCL91, OCZ94, BC95, Tay96]. The first approach is based on traditional FV methods [Pat80] as applied to problems in CFD and suffers from the same difficulties when applied to complex geometries involving arbitrarily structured meshes [DM92, HH95]. The second approach is based on traditional
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FE methods [ZT89] and employs shape functions to describe the variation of a variable over an element, and is therefore well suited to complex geometries [FBCL91, OCZ94]. Both approaches apply strict conservation over a control volume and have demonstrated superiority over traditional FE methods with regard to accuracy [Whe96, Tay96], some researchers have attributed this to the local conservation of a variable as enforced by the control volumes employed [FBCL91, BC95] and others have attributed it to the enforced continuity of the derivatives of variables across cell boundaries [Whe96]. The objective of this paper is to describe the application of a vertex-based FV method to problems involving elasto-plastic deformation and provide a detailed comparison with a standard Galerkin FE method. 2. Equilibrium Equations and Boundary Conditions
In matrix form, the incremental equilibrium equations are where [L] is the differential operator, {Acr} is the Cauchy stress, {&} is the body force and n, is is the domain. The boundary conditions on the surface T = Tt U Fu of the domain ft can be defined as [ZT89, OCZ94]
where {tp} are the prescribed tractions on the boundary Ft, {up} are the prescribed displacements on the boundary Tu and [R] is the outward normal operator [OCZ94, Tay96]. 3. Constitutive Relationship
In matrix form, the stress is related to the elastic strain incrementally as follows; {Acr} = [_D]{Aee}, where [D] is the elasticity matrix. For the deformation of metals, the von-Mises yield criterion is employed and the elastic strain is given by {A6e} = {Ae} — {Aevp}, where {Ae} and {Ae^p} are the total and visco-plastic strain, respectively. The visco-plastic strain rate is given by the Perzyna model [Per66]
where aeq, ay, 7, N and s are the equivalent stress, yield stress, fluidity, strain rate sensitivity parameter and deviatoric stress, respectively. The < x > operator is defined as follows;
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The total infinitesimal strain is {Ae} = [L]{Aw}, where {Aw} is the incremental displacement.
4. Vertex-based Discretisation Employing the constitutive relationship of the previous section in equations (1) and ( 2 ) , and assuming the boundary conditions as described by equation (3) are directly satisfied by the vector {Aw}, the method of weighted residuals can be applied to the equations to obtain the following weak form of the equilibrium equation [ZT89];
where [W] is a diagonal matrix of arbitrary weighting functions. At this point the unknown displacement can be approximated as
where {Au}j is the unknown displacement at the vertex j, Nj is the shape function associated with the unknown displacement and [/] is the identity matrix. The displacement approximation can be introduced into equation (5) if the arbitrary weighting functions [W] are replaced by a finite set of prescribed functions [W] = £"=1[^],, for each vertex i [ZT89, OCZ94], /»
f
Equation (7) can be expressed as an incremental linear system of equations of the form [A'] {Aw} — {/} = {0}, where [K'] is the global stiffness matrix, {Aw} is the global displacement approximation and {/} is the global equivalent force vector and can be formed from the summation of the following contributions;
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Figure 1: 2D control volumes, (a) overlapping FE and (b) non-overlapping FV. where £7; is the control volume associated with the vertex i and F; = F U i U Tti is the boundary of the control volume.
4.1. Standard Galerkm FE Method In the standard Galerkin FE method the weighting function associated with a vertex is equal to the shape function of the unknown associated with tha' vertex [ZT89, Hir88, OCZ94], [W], = [N]i. The shape functions describe the variation of an unknown over an element and there can be a number of elements associated with each vertex. Hence, it is apparent that control volume: described by weighting functions of this form will always overlap. This is illus trated in Figure l(a) for a simple two dimensional case of two adjacent node; i and j, where the control volumes fij and nj have contributions from all the elements associated with their respective vertices i and j. Hence, for the standard Galerkin FE method the contributions as describee by equations (8) and (9) are
where [B],- = [LN]i. It is important to note that if the boundary of the control volume, such as that described by F; in Figure l(a), coincides with the external boundary of the domain, the shape functions are not necessarily zero along that part of the boundary. Thus, if a flux is prescribed such as a traction this will not necessarily disappear and may contribute to the equivalent force vector as described in equation (11). Additionally, the symmetrical nature of the stiffness matrix as indicated by equation (10) should be noted. The Galerkin approach is
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accepted as the optimum technique for treating physical situations described by self-adjoint differential equations, particularly those in solid mechanics, as the inherent symmetrical nature is preserved by the choice of weighting functions [ZT89, OCZ94]. 4.2. Vertex-based FV Method In the vertex-based FV method the weighting functions associated with a vertex are equal to unity within the control volume, [W]i = [/], and zero elsewhere. This definition is equivalent to that for the subdomain collocation method as defined in the standard texts [Hir88, ZT89]. Though it is important to note that weighting functions defined in this manner permit a variety of possibilities with regard to the control volume definition [OCZ94]. This is because the weighting functions are not restricted to to a direct association with the cell or element as in the Galerkin case. This is an important consideration and requires the recognition of the vertex-based FV method as a discretisation technique in its own right [Hir88]. For the vertex-based FV method the contributions as described by equations (8) and (9) are
It is important to note that the traction boundary conditions can be applied directly as another surface integral, but in the previous Galerkin approach an additional surface element is generally included on the domain boundary. A non-overlapping control volume definition suitable for a vertex-based FV method is illustrated in two and three dimensions in Figures l(b) and 2(a), respectively. The Figures illustrate the assembling of vertex-based control volumes from their required sub-control volumes [Tay96]. Additionally, the asymmetric nature of the contributions to the overall stiffness matrix as described by equation (12) does not ensure that symmetry will always be preserved. For this reason FV methods were initially argued as being inferior, but in the light of recent research where different control volume definitions have been proposed, the extent of this inferiority has come into question [OCZ94, Zie95, BC95]. 5. Results and Conclusions
In this section the vertex-based FV method is applied to a three dimensional validation problem and compared with the standard Galerkin FE method. The non-linear solution procedure adopted in for both these methods is based upon
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Figure 2: (a) 3D assembly of FV sub-control volumes and (b) spherical vessel. that of Zienkiewicz and Cormeau [ZC74, Tay96]. Both methods utilised an explicit technique with regard to time stepping of the Perzyna equation (4). It is important to note that the FV solution procedure only differs from that of the FE in contributions to the global equivalent force vector and the global stiffness matrix. Hence, allowing an accurate comparison of the two methods [Tay96]. The methods are compared with regard to accuracy and computational cost. They are also analysed for a variety of meshes with different element assemblies. 5.1. Test case: Internally pressurised spherical vessel For this validation problem a thick walled spherical vessel, consisting of an elastic-perfectly plastic material, undergoes an instantaneously applied internal pressure load. The pressure load is 30 dNmrn" 2 , the Youngs modulus and Poisson ratio required to define the elasticity matrix are 21,000 d N m m - 2 and 0.3, respectively, and the yield stress is 24 dNmm- 2 . This problem is rate independent and the final solution is equivalent to that of an elasto-plastic analysis [ZC74]. A closed form radial solution is available [Hil50]. Numerically the problem can be modelled in three dimensional Cartesian coordinates, with the displacement components fixed to zero in the relative symmetry planes. The spherical vessel is then reduced to an octant as illustrated in Figure 2(b) 1 . Examples of meshes consisting of linear tetrahedral (LT), bilinear pentahedral (BLP) and trilinear hexahedral (TLH) elements are illustrated in Figures 2(b) 2 , 2(b) 3 and 2(b) 4 , respectively. Firstly, the problem was analysed with a series of meshes consisting of TLH elements. The hoop stress profiles, along the radii, as obtained from one of the numerical analyses are plotted and compared against the reference solution in Figure 3(a). The profiles illustrate the stress in the plastic and elastic regions, and the radial extent of the plastic region according to the analytical solution. The close agreement of the two methods is illustrated. However, it
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Figure 3: (a) 950 TLH and (b) 4,800 LT elements. is important to note the closer agreement between the reference solution and the FV method when a coarse mesh is employed. These observations may be associated with the higher order, trilinear nature of the elements employed in the three dimensional analysis at this stage. With regard to the FV method, the implementation of pressure loads (tractions) will involve bilinear face elements for TLH elements. Hence, when considering the application of pressure loads for the two methods as described in equations (11) and (13), the contributions are different due to the individual weighting technique associated with each method. Furthermore, the weighting technique employed for the FV method may be more complementary, when applied generally, as all the terms are integrated conservatively at a local level. Conversely, for the FE method the weighting is not locally conservative which may introduce errors when pressure loads are employed. These conclusions are tentative and rely on the interpretation of the present observations, but they agree with the findings of other researchers [Whe96] and strongly suggest that further research of the FV method is worthwhile. Secondly, the problem was analysed with a series of meshes consisting of BLP elements and there was much closer agreement between the methods [Tay96]. This is attributable to the lower order, bilinear nature of the element concerned and the linear nature of the triangular faces over which the pressure loads were applied. As illustrated in Figure 2(b) 3 the BLP elements are orientated so the pressure load was prescribed over a triangular face. This was an outcome of the automatic mesh generator employed [Fern] and it is possible to further study the element when pressures are applied to the bilinear, quadrilateral faces, though it was not studied in that research. Thirdly, the problem was analysed with a series of meshes consisting of LT elements. The hoop stress profiles from one of the analyses are plotted in Figure 3(b). There is complete agreement between the methods with regard to LT elements as the global stiffness matrices and global force vectors constructed by the two methods are identical. This is a consequence of the linear nature of
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Figure 4: (a) CPU times on a SPARC 4, 110MHz. both the element concerned and the triangular faces over which the pressure is applied. It is possible to demonstrate this equivalence analytically [Tay96] by extending to three dimensions, a two dimensional analysis which has been applied to elastic problems involving linear triangular elements [OCZ94]. Finally, the methods were compared with regard to computational cost. Considering LT elements, as the matrices are identical and symmetric a conjugate gradient method (CGM) is applicable in both cases. As illustrated in Figure 4(b), the FV method (FV-CGM) requires more CPU time than the FE method (FE-CGM) even when the same linear solver is employed. This is expected as the FV method visits six integration points, while the FE method visits a single Gauss point when adding contributions to the linear system of equations [Tay96]. Considering TLH elements, the geometrical nature of this validation problem prohibits an orthogonally assembled mesh. Hence, for the FV method a bi-conjugate gradient method (Bi-CGM) is required due to the asymmetric nature of the coefficient matrix obtained [Tay96j. Conversely, for the FE method a CGM is sufficient as the matrix obtained is symmetric. These requirements agree with the discussions in the previous section. As illustrated in Figure 4(a), the FV method (FV-BiCGM) requires approximately twice the CPU time as the FE method (FE-CGM). This is also expected due to the computational requirements of the two different linear solvers employed. Also for TLH elements, the FV method visits twelve integration points per element, while the FE method visits eight Gauss points per element. Hence, it can finally be concluded that any improvement in accuracy obtained by employing the vertex-based FV method must be offset against the extra computational cost required. Bibliography [BC95]
C. Bailey and M. Cross. A finite volume procedure to solve elastic solid
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mechanics problems in three dimensions on an unstructured mesh. Int. Journal for Num. Methods in Engg., 38:1757-1776, 1995. [DM92]
I. Demirdzic and D. Martinovic. Finite volume method for thermo-elastoplastic stress analysis. Computer Methods in Applied Mechanics and Engineering, 109:331-349, 1992.
[FBCL91] Y.D. Fryer, C. Bailey, M. Cross, and C.-H. Lai. A control volume procedure for solving the elastic stress-strain equations on an unstructured mesh. Appl. Math. Modelling, 15:639-645, 1991. [Fern]
Femview Ltd., Leicester, UK.
FEMGEN/FEMVIEW.
[HH95]
J.H. Hattel and P.N. Hansen. A control volume-based finite difference method for solving the equilibrium equations in terms of displacements. Appl. Math. Modelling, 19:210-243, 1995.
[HilSO]
R. Hill. The Mathematical Theory of Plasticity. Clarendon Press, Oxford, UK, 1950.
[Hir88]
C. Hirsch. Numerical Computation of Internal and External Flows: Fundamentals of Numerical Discretisation, volume 1. John Wiley and Sons, 1988.
[OCZ94]
E. Onate, M. Cervera, and O.C. Zienkiewicz. A finite volume format for structural mechanics. Int. Journal for Num. Methods in Engg., 37:181-201, 1994.
[OH80]
D.R.J. Owen and E. Hinton. Finite Elements in Plasticity: Theory and Practice. Pineridge Press Ltd., Swansea, UK, 1980.
[PatSO]
S.V. Patanker. Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington DC, 1980.
[Per66]
P. Perzyna. Fundamental problems in visco-plasticity. Advan. Appl. Mech., 9:243-377, 1966.
[Tay96]
G.A. Taylor. A Vertex Based Discretisation Scheme Applied to Material Non-linearity within a Multi-physics Finite Volume Framework. PhD thesis, The University of Greenwich, 1996.
[Whe96]
M.A. Wheel. A geometrically versatile finite volume formulation for plane elastostatic stress analysis. Journal of strain analysis, 31(2):111-116, 1996.
[Whe99]
M.A. Wheel. A mixed finite volume formulation for determining the small strain deformation of incompressible materials. Int. Journal for Num. Methods in Engg., 44:1843-1861, 1999.
[ZC74]
O.C. Zienkiewicz and I.C. Cormeau. Visco-plasticity—plasticity and creep in elastic solids—a unified numerical solution approach. Int. Journal for Num. Methods in Engg., 8:821-845, 1974.
[Zie95]
O.C. Zienkiewicz. Origins, milestones and directions of the finite element method - a personal view. Archives of computational methods in Engg., 2:1-48, 1995.
[ZT89]
O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method: Volume 1: Basic Formulation and Linear Problems. Magraw-Hill, Maidenhead, Berkshire, UK, 1989.
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Control volumes technique applied to gas dynamical problems in underground mines
Elena Vlasseva Associated Professor Department of Mine Ventilation and Labour Safety University ofMining&Geology "St. Ivan Rilski" Sofia 1100, Bulgaria
ABSTRACT. Paper presents an application of Control Volumes' Method in the field of underground mining. Modeling deals with noxious gas distribution (concentration) in time and space along mine roadway or network. Models take into account variable velocity, density and outside gas inflows in mass balance of air flow. Different source functions, which due to technological or accidental reasons, provoke occurrence of transient process, are presented in the paper. Numerical and computer models are verified with real data, obtained unfortunately from great disaster, which took place in one Bulgarian underground coal mine in 1997. Models serve as a tool used to reveal the circumstances lead to that disaster. Modeling results might be also applied to help introduction into practice of engineering solution, which can be analyzed due to their practical implementation. Key words: control volumes; gas distribution; underground mining
1.
Introduction
Air flows in underground mines are of great importance for human lives underground. Normally flows transfer not only fresh air but different impurities, liberated from rocks or as a result of mining operations. Keeping of these impurities into safety limits is one of the main task for underground staff. Once liberated into underground air, gas is distributed along ventilation paths in difficult to predict ways and time duration. Gas might be observed on unlikely for common understanding places. By that reason preliminary knowledge about gas dynamics is extremely valuable. One possible approach might be real mine experiments. This approach however faced great difficulties due to problems with simulation of realistic gas liberation characteristics. Problems with control of such simulation cannot be neglected as well. All the above points show that mathematical and computer
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modeling can be of favor in prediction of gas distribution along ventilation paths. They can reveal specific relationship between gas liberation and its influence on underground air flows.difficulties with control and safety of the experiment. In that connection mathematical and computer modeling in close with reality decsription is very important and liable technique. It can reveal specific relationships between gas source and its influence on ventilation system and its important parameters. The process of gas liberation and distribution in a single path and its mixing with fresh air is presented by convection-diffusion equation with source function, varaible velocity and density of air gas mixture. It is solved simultaneously with continuity equation. Control volumes method with exponential scheme is applied for numerical modeling.. Errors due to discterization and limits of method for this class of problems are outlined. Solutions for one ventilation path are addapted to a network modeling where interrelation between air flows with different characteristics are of a great importance to the whole process of safety working conditions in mine. Results of modeling are validated with practice in the following ways: ^ it was used as investigation tool for one methane explosion which took place two years ago in Bulgaria. Revealing of most likely reasons for its occurrence became possible; > it was also applied to in one case study for inertization with nitrogen of a typical mine configuration. This technical activity is performed to supress mine fire development. 1.
Governing equations and source functions
Let us assume mine roadway with characteristics shown on figure 1. Air flow is well developed in direction of coordinate s. Length of roadway (L) is times greater compared to its width or cross sectional area (f) and this give a ground to present mathematical model into one dimensional way. Independent variables are: • s - along the length of mine roadway, m; • 1 - time for tracing the process, s.
Figure 1. Physical model.
Figure 3. Discretization of calculation area
Known functions are: U(S,T) - velocity of air gas mixture, m/s;
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q(s,i) - gas liberation, m3/s; P(S,T) - density of air gas mixture, kg/m3. Unknown function is C(S,T) - concentration of gas impurity into air gas mixture. Gas distribution in thus describe physical model obeys mass conservation for gas impurity and air and equation of state for air gas mixture: • •
In the above written equations index g referred to gas impurity, index m - to air gas mixture. Initial and boundary conditions are given with the terms:
Source functions q(s,T) which present gas liberation or provocation of transient process are presented on figure 2. Their characteristics may change either in time or in space or combinations of both. They are chosen on typical behavior of gas sources: a - gas liberation from walls or during coal transportation; b, c - gas liberation from already mined areas; d - point gas source; e - gas liberation due to repeatable operations (diesel power, blast work); f - gas liberation from mined areas with increase of its surface; g - gas desorption from newly opened surfaces. Figure 2. functions
Typical
source
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Finite volumes for complex applications
Combinations of above presented source functions are also possible. For istance: a + d - point gas source with constant emission rate in time; c + f - distributed along the length source with variable in time emission rate; b + g - linearly distributed source along the length with pick in time and then gradually decreasing emission rate; d + e - point source with periodically changeable emission rate. Presented on figure 2 source functions can be easily approximated with polinom or with sum of functions. Numerical treatment of above presented mathematical model - expressions (1-3) plus (4-5) makes possible taking into account all written into model variations. 2.
Numerical and computer modeling
2.1. Schemes, mesh-type, discretization Mathematical model (1-5) is solved by application of Control Volumes Method [PAT84]. Presented on fig. 1 physical model is transformed into regular mesh of control volumes (fig. 2) with size As. They are defined by mesh points and control volumes boundaries. Number of points are Concentration
C(Si,Tj)
is
defined
in
mesh
points,
air-gas
flows
Exponential profile [PAT84} for concentration variations is assumed. Following the above points numerical analogue, binding three adjacent points (W,P,E) can be written in the way:
where:
Upper index 0 referred to a previous time step. Boundary conditions (4-5) are transformed into numerical schemes in the same way by assuming first order boundary condition at left boundary (s=0) and second order - at the right boundary.
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Full procedure on transformation of differential problem into numerical schemes is given in [VLA93]. Having expressions (7) for each three adjacent points linear system with three diagonal matrix for all points i d ( l - N x ) i s obtained. Its solution under TDMA gives concentration in each mesh point and in any time moment C(SJ,TJ). 2.2. Approximation, stability, convergence, errors, application limits Numerical schemes (7) approximate differential problem (1-5). This statement was proved by application of Taylor's series. Approximation error is from first order in regard of 1 and from second order in regard of s. Numerical scheme (7) is absolutely stable. It was proved by applying Matrix criterion [SHI88] by presenting numerical scheme in way from one time layer to the next one and examining eigen vectors and their eigen values [SHI88]. Convergence is a concequence from approximation and stability, following Lax theorem [SHI88]. Applicability limits to gas dynamical problems are investigated. For most common parameters - velocities, lengths etc. error analysis was performed. On figure 4 relative errors (numerical/exact solution) in regard of
^
Cu number (Cu =
Figure 4. Relative errors due to discretization
3.
uAT
As
). Number of
mesh points, even limited to the minimal range (3) lead to 8% errors, which for purpose of gas dynamical problems is agreeable. Physically unlikely results has not been obtained.
Modeling validation with practice
Presented herewith mathematical model and its numerical interpretation reach their computer realization in both aspects: • for a single roadway or set of sequential ones [VLA93]; • for a complex network, where solutions for a single roadway were harmonized at places where they cross each other [STE87]. Modeling was applied in numerous engineering problems from practice. Two of them are presented in this paper: • investigation of one methane explosion; • technological solutions for inertization with nitrogen of fire zone. 3.1. Methane explosion
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Finite volumes for complex applications
Severe methane explosion took place on the 1-st of September 1997 in "Ivan Russev" coal mine. Operations in one section were canceled due to annual holidays from 1st August to 1st September. As a result great amounts of methane were accumulated. Degazation (transfer of this gas out of mine) was to be performed. Mine does not have remote control system and no data (besides miners' evidences) was available. As a result of incorrect activities of miners, performing degazation, explosion took place causing death for more than 10 workers.
Figure 5. Path in mine for distribution of methane during degasing Investigation about circumstances [MIC98] was performed by application of models, presented in this paper. Investigated sector of mine is shown on figure 5.
Figure 7. Concetration at the entrance of Figure 6. Air volumes at the entrance of observed path observed path Changes in cross sections can be seen there. Transient process was initiated by variable air flow (figure 6) and methane concentration (figure 7) at starting point 0. They reflect workers actions, such as: • variable cross-sections along the route (fig. 11); • assembling of ventilation curtain to direct higher air flow towards the gassed section; • switching on/off of booster's fan operation, causing changes in ventilation conditions (air volumes Q 1 Q 2
and
Q3
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variable methane concentration (C2) of in-flowing air into 503 crosscut (Q2)
Explosion took place between points 5 and 6 and modeling results (figure 8) show the same - high methane concentration with more than 7 minutes duration stay at the place of explosion. Unfortunately ignition source was also available. Figure 8. Concetration in time of observation in some points of path
3.2. Inerting with nitrogen of fire zone In some cases during mine operations evidence for [MIC97] development of mine fire can be observed. Then one of anti-fire measures is to inertize atmosphere at a danger zone so that oxygen does not be available to support burning. The model presented herewith can deal with one impurity into air-gas mixture. Inerting of air however presumes more than one ipurity, namely: • methane inflow from mined zone (points 5-6 on figure 9) and from mine workings (points 1-2-3-4 on figure 9); • oxygen from ventilation flow and from injected technical nitrogen; • nitrogen from air and from injected technical nitrogen. In order to evaluate concentration of the above mentioned three gas components via a model constructed for a single component, the author has applied consecutive diffusion mixing. It pressumes appropriate definition of transporting medium and impurity as well as suitable presentation of gas sources q(s, T) . Calculational passes three stages (methane release and distribution, nitrogen outflow from gob area, nitrogen injection at a given place in the panel and its further distribution in the area which must be inert). For any of these stages computer modulus were developed - METHANE, INERT_GOB and NITRO. Common input data for the three computer programs are geometric characteristics of mine workings (fig.9). The three programs interact and their incorporating in the total inverting strategy makes it possible the composition of general program procedure INERTIZATION Numerous solutions were performed, corresponding to specific fire situations. Very important parameters were obtained: • time needed to inert the observed object; concentration of flammable gas in all points of observed area (this Figure 9. Object for inertization
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Finite volumes for complex applications
information is very important in order to keep the atmosphere out of explosion); • effectiveness of operations performed on fire supression. On figure 10 is shown methane concentration along the path with length 1500 m. Figure 11 shows time delay in inertization from point 2 and 3.
Figure 10. Time delay in inertization
Figure 10. Methane concentration
4. Bibliography [VLA93]
[MIC98]
[MIC97]
[PAT84] [SHIS8] [STE87}
Vlasseva E.D., Mathematical Modeling of Convection-diffusion Processes in Underground Mines, Ph.D. Thesis, UMG, Sofia, 1993, pp. 200 Michaylov M.A., E.D.Vlasseva, Simulation Analysis of Methane Explosion, Second International Symposium on Mine Environmental Engineering, July 29-31 1998, Brunei University, UK,p. 1-16. Michaylov M.A., E.D.Vlasseva, Modeling of Preventive Nitrogen Inertization in Underground Mines, 15th Mining Congress of Turkey, 6-9 May 1997, Ankara, pp. 203-210. Patankar S., Numerical Methods in Heat Transfer, Sovremennoe Mashinostroene, Moskow, 1984, p. 150 (Russian translation) Shi D., Numerical methods in Het Transfer Problems, Moskow, Mir, 1988, p. 544 (Russian translation) Stefanov,T.P., E.D.Vlasseva, E.E.Arsenyan, Unsteady Gas Flows in Mine Ventilation Networks, 22 International Conference of Safety in Mine Research Institutes, Beijing, China, oct.1987,pp. 115-124.
Simulation of salt-fresh water interface in coastal aquifers using a finite volume scheme on unstructured meshes B. Bouzouf, D. Ouazar LASH, EMI 14 Av. Ibnsina Rabat, Maroc I. Elmahi IVG, University of Duisburg, Germany
ABSTRACT This paper is devoted to the numerical study of seawater intrusion into coastal aquifers. The cell-centered finite volume method is adopted here to solve the set of simultaneous partial differential equations describing the motion of saltwater and freshwater separated by a sharp interface. These equations are based on the Dupuit approximation and are obtained from integration over the vertical dimension. In order to have flexibility upon complex configurations domain, non structured grid meshing is utilized. To approximate the diffusion fluxes, GreenGauss type reconstruction, based on Diamond cell and least square interpolation, is performed. The model is first validated using academic test case studies with known close from solutions. A real case study concerning the Gharb aquifer in North West of Morocco is carried out to show the overall trend of saltwater fronts. Key Words: Coastal aquifers, seawater intrusion, finite volumes, unstructured meshes, Green-Gauss reconstruction. 1. Introduction The modelling of groundwater in coastal aquifers is an important and difficult issue in water resources. The primary difficulty resides in efficient and accurate simulation of the movement of the saltwater front. Freshwater and saltwater are miscible fluids and therefore the zone separating them takes the form of a transition zone caused by hydrodynamic dispersion. For certain problems where the transition zone is relatively small compared to the aquifer extent and thickness, the simulation can be simplified by assuming that two fluids are immiscible and separated by a sharp interface (sharp interface model). This last assumption, together with the Dupuit approximation, permits the integration of the equations in the vertical direction [BEA 79]. The objective of this paper is to present a cell-centered finite volume based approximation to calculate the position of the sharp interface. This class of methods
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Finite volumes for complex applications
is becoming one of the commonly used techniques for partial differential equations in engineering calculations and computational physics. Their popularity is due mainly to their ability to faithful to the physics conservation and the possibility of solving the problems on complex geometries. The diffusion contributions here are approximated by using Green-Gauss type interpolation. This technique is found to be very robust, it can be used on general mesh, not satisfying necessarily Delaunay condition on the triangulation. Time integration is performed by an explicit Euler scheme in order to keep the memory requirement reasonable. 2. Mathematical Model
We assume here that the saltwater and freshwater are separated by a sharp interface, thus two domains are considered. For each flow domain the equation of continuity may be integrated over the vertical dimension reducing the determination of the position of the interface to a 2D problem. The system of equations can be written as follow ([ESS 90]):
Qf and ns represent here the fresh and salt water flow domain respectively, Kfx and KSx (respectively Kfy and Ksy ) are the hydraulic permeabilities in the fresh and salt water in x-direction (respectively in y-direction), hf and hs are the heads, Bf and Bs are the thickness of fresh and salt water zone and n is the porosity. We note also by 6 = —, where pf and ps are the specific weights in fresh Ps - Pf and salt water, and by 0 for confined aquifer 1 for unconfined aquifer
(
Invoking continuity of the pressure at the interfaces, the interface elevation can be calculated from the freshwater and saltwater heads by
The system (1) represents two coupled parabolic partial differential equations which should be solved simultaneously for the freshwater head (hf) saltwater head (hs). Once these values are known, the interface elevation (£) can be obtained from (2). The set of the equations (1) can be written in the vectorial form:
Fields of application
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3. Finite volume discretization
To solve the system of equations (3) we have considered a triangular cellcentered finite volume formulation ([EBVGPH 99]), where the state variables W™ are the average values for the cells at time level n:
Integrating eq. (3) on a control volume d yields in explicit formulation:
where Ai is the time step. The discretization of equation (4) requires the approximation of terms such as
where / = /, s and F^ is the interface separating two cells d and Cj. Y. Coudiere et al. [CVV 96], have studied an elliptic problem
Where A is a symmetric definite positive matrix with coefficients aij in C1(il), / € C°(fy and g € C2(T). They have used a Green-Gauss type interpolation to construct the gradients at the interfaces of the mesh. The gradient on each edge is approached by the Green theorem and then a first order Gauss quadrature formula, for which requisite values at the vertices P are interpolated from the states on the neighbourhood of P. The weak consistency of this scheme was
528
Finite volumes for complex applications
proved under some assumption on the weights of the interpolation. We took inspiration from this scheme for devising our numerical procedure and discretize the diffusive contributions. We begin firstly by writing
One constructs the co-volume Cdec centered at the interface Fij and connecting the barycenters Gi and Gj of the triangles that share this edge and the two endpoints N and S of FJJ (see figure 1).
Figure 1: Diamond shaped co-volume or
To calculate -^— | r • • , the divergence theorem is applied to the co-volume Cde ox surrounding Fij, which gives
e represents an edge of the co-volume Cdec and nxe is the axial component of the outward unit normal to e. If we note by e — [N1, N2], One can write also
Where hi N1 and hi N2 are respectively the values of hi on the nodes N1 and N2 of the edge e. The data at the centers d and Gj are known exactly while the data at the vertices N and 5 must be determined by some interpolation procedure. For one node P
Fields of application
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of the mesh, one utilizes a linear approximation v of hi on the set of cells which share the vertex P,
Where V(P) is the set of triangles K surrounding P, hiK the head at the center of triangle K and CCK(P) are the weights of the interpolation corresponding to the node P. In order to ensure the consistency of the scheme, the weights of the interpolation are calculated using a least square procedure (see [CVV 96] for details). 4. Model validation
To verify and validate the numerical solution obtained from the finite volume model, numerical simulations have been compared to existing analytical solutions. 4.1 Steady state Two cases have been checked: confined and unconfined aquifer. For both of them the initial values of hf and £ are arbitrarily fixed. The analytical solutions are as follows: . Unconfined aquifer ([VER 68], fVN 751):
with
and
• Confined aquifer ([GLO 59},{RH 62]):
with 4.2 Unsteady state Keulegan [KEU 54] presented an analytical solution for the interface in a confined aquifer of uniform thickness:
with D = 10m , n = 0.3 and K = 39.024m/day.
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Finite volumes for complex applications
Figure 2: Comparison of analytical and numerical solutions for unconfined aquifer.
Figure 3: Comparison of analytical and numerical solutions for confined aquifer.
Figure 4: Comparison between analytical solution and finite volume method.
Fields of application
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The numerical solutions are in good agreement with the analytical solutions as depicted in figures (2), (3) and (4). 5. Application to the Gharb basin, Morocco
We have applied the finite volume based model to the Gharb aquifer which is located in North West of Morocco. The surface area of the coastal Gharb is estimated to about 4000 km 2 . It belongs to the structural domain of the Morocco Atlantic plain.
Location of the Gharb basin In figure (5), the areas of pumping in the Gharb aquifer are depicted. Figure (6) shows the actual front saltwater corresponding to pumping schemes of Figure (5). 6. Conclusion
Characterization of certain coastal aquifer systems may be accomplished by assuming that saltwater and freshwater are separated by a sharp interface. Invoking the Dupuit assumption and performing a vertical integration results in quasi-three-dimensional, the equations may be solved to give freshwater head, saltwater head and interface elevation. Cell-centered finite volume scheme on a unstructured mesh is used to approximate the partial differential equations. Comparisons of the finite volume approach adopted in this paper, with known analytical solutions have shown close agreement. The model was also applied to a real case concerning the Gharb aquifer in North West of Morocco.
BIBLIOGRAPHY [BEA 79]
BEAR J., Hydraulics of groundwater, McGraws-Hill, New York, 569 pages, 1979.
[CVV 96]
COUDIERE Y., VILA J. P. AND VILLEDIEU P., Convergence of a finite volume scheme for a diffusion problem, F. Benkhaldoun
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Finite volumes for complex applications
Fig. 5 Areas of pumping in GHARB Fig. 6 Simulated freshwater and saltwater aquifer interface
and R. Vilsmeier eds, Finite volume for complex applications (Hermes, Paris), pp. 161-168, 1996. [EBVGPH 99]
ELMAHI I., BENKHALDOUN F., VILSMEIER R., GLOTH O., PATSCHULL A. AND HANEL D., Finite volume simulation of a droplet flame ignition on unstructured meshes, J. of Comput. and Appl. Math., Vol 103, 1, pp. 187-205, 1999.
[ESS 90]
ESSAID H. I., A quasi-three-dimensional finite difference model to simulate freshwater and saltwater flow in layered coastal aquifer systems, U.S. Geological survey Water-Resources Investigations, Report 90-4130. Menlo Park, California, 1990.
[GLO 59]
GLOVER R. E., The pattern of freshwater flow in a coastal aquifer, J. of Ground Water Resour., 64, pp. 439-475, 1959.
[KEU 54]
KEULEGAN H. G., An example report on model laws for density current, U.S. Natl. Bur. of Stand., Gaitherburg, Md, 1954.
[RH 62]
RUMMER R. R. AND HARLEMAN D. R., Intruded saltwater wedge in porous media, U.S. Geol. Surf. Prof., Paper 450-B, 1962.
[VER 68]
VERUIJT A., A note on the Ghiben-Herzberg formula, IASH bull. 13, pp. 43-45, 1968.
[VN 75]
VAPPICHA V. N. AND NAGARAJA S. H., Steady state interface in coastal aquifer with a vertical outflow face, National Symposium on Hydrology, Rurkee, India, 1975.
Progress in the flow simulation of high voltage circuit breakers X. Ye, L. Miiller, K. Kaltenegger and J. Stechbarth ABB High Voltage Technologies Ltd., 5401 Baden, Switzerland ABSTRACT In this paper progresses in the physical and numerical modelling, which lead to improvement in the accuracy and capability of simulation for capacitive switching design of circuit breakers, are introduced. Numerical results and measurement results are compared and discussed. One important progress lies in the successful treatment of the artificial viscosity. To maintain the numerical stability but at the same time to keep the artificial viscosity so small that the physical viscosity is not distorted, the upwind biasing essentially local extremum diminishing (ELED) scheme has been adopted and improved. Another progress is associated the moving grids technique, where additional terms have been added to the governing equations for the moving grid without deformation and new grid lines are added or removed in the deformed moving grid by the solver in the progress of the calculation. The Chimera boundary interpolation method is used to enable the communication between two blocks with relative motion. Key Words: circuit breaker, moving grids, upwind biasing scheme, Chimera boundary
\. Introduction In a high voltage circuit breaker a gas with good dielectric and thermodynamic properties such as SF6 is used to extinguish the electric arcing which occurs as electric contacts move apart. The current interruption can only then be realised. The capacitive switching, i.e., the current interruption at high voltage but with low current, represents one important case among various cases of tests and design. The ability to perform capacitive switching without electric breakdown is one of the defining parameters for the speed of the circuit breaker and therefore strongly cost relevant. Furthermore, only the density of gas (p) and its electric field strength (£) play an important role in gas breakdown between the contacts of the circuit breaker (s. Fig. 4). As the criterion of the ratio ofE/p for gas breakdown is well known, CFD can be employed to simulate the flow field and to subsequently produce the distribution of gas density in the circuit breaker during the design. It is obvious that for such simulations, a high degree of accuracy of flow calculation is required.
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Finite volumes for complex applications
Since later 1980's researchers and engineers have begun to apply CFD tools in the development of circuit breakers. Most of them, e.g. [CLA 97], concentrated their efforts on using the flow simulation with implemented arcing models to investigate the ground flow effects in circuit breakers, e.g., pressure build-up in pressure chamber, plasma jets, and to identify the limit and capability of CFD tools. Their efforts to verify CFD tools were, however, constrained mainly in the comparison of the simulated pressure build-up in the pressure chamber with experiments. There are only few works (s. [TRE 91]) done for enhancing the accuracy of CFD tools for predicting dielectric strength in capacitive switching and for verification of CFD tools in the regions of nozzles and electrode contacts. As a result, the ability and accuracy to predict the dielectric strength of a circuit breaker with CFD tools have been not satisfactory. The following two points remain unclear. 1) How can the moving parts be treated correctly and which numerical methods are to be introduce to obtain a sufficient high accuracy? 2) How can a CFD tool be verified for predicting dielectric strength in capacitive switching? In this paper, these questions are addressed, with concentration on the simulation of cold SF6 gas with moving electric contacts and further on the prediction of the dielectric strength of a circuit breaker during the capacitive switching. The cold gas simulation is not as trivial as to be anticipated, because the low energy level of cold gas forces a CFD code to consider, with sufficient accuracy, all macroscopic and microscopic flow effects, examples of which are: flow separation, influence of wall and viscose layer and their transient evolution, turbulence transport and suck effect caused by moving contacts. Further, to guarantee the numerical stability, artificial viscosity must be introduced, and hence a sophisticated scheme must be used to keep the artificial viscosity so small that it can maintain the numerical stability effectively but does not confuse the physical viscosity on a viscose layer. A sophisticated moving grids technique has to be introduced both for obtaining a high accuracy and for performing an efficient computation. In section 2, the mathematical models and numerical methods, such as ELED scheme, moving grids technique and Chimera boundary, are introduced. In section 3, examples are presented and discussed for flow simulation where the code verified through the comparison with experiment. Subsequently, the method and example are introduced for coupling of the flow field and the electric-static field in circuit breakers with the consideration of the influence of roughness. Finally, in section 4, our results are concluded with further improvement suggested.
2. Mathematical models and numerical methods
2.1 Mathematical models For the fluid flow in a circuit breaker it is necessary to use the complete NavierStokes equations in their time averaged form, i.e. the so called Reynolds equations
Fields of application
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with an adequate turbulence model. The fluid flow will be then governed by the Navier-Stokes conservation equations for mass, momentum and energy. These equations have the differential form as shown in eq. [1].
with the variables
with the shear stress I and heat flux Jas follows:
where ///, //, are molecular and turbulent viscosity, A heat conduction coefficient, 0 temperature, Pr Prandtl number, V Nabla operator, Re Reynolds number, W velocity vector. These equations can be transformed to curvilinear coordinate system with the transformation <^=^(x,y) and T]=T](x,y). To close the equation system, the standard k-£ turbulence model (s. [LAU 74]) was used to obtain the turbulent viscosity ju,.
2.2 Numerical methods The basic equations are solved with a multi-block finite volume Runge-Kutta multisteps time-marching method. The code (HT206) was previously applied for fluid flow in turbo systems (s. [SCH 91] and [SCH 98]) and has been extended for the development of circuit breaker. The numerical methods of the code, which are relevant to the flow simulation of circuit breaker, are described as follows: 2.2.1 ELED scheme The spatial discretisation of the present code is based on a high order non-oscillatory scheme, which consisting of central discretisation and artificial viscosity in the following form (s. [JAM 81]):
536
Finite volumes for complex applications £{ '
where q denotes the variables in the equation [1] and ,.i and "y+i are the ;4 coefficients of the scheme. If the formulation of these coefficients is taken from [JAM 81], it is then the classic Jameson-Schmidt-Turkel (1ST) scheme. A more advanced formulation of these coefficients is introduced in the present code, that is the essentially local extremum diminishing (ELED) scheme of [JAM 94]:
The valuation of the wave speed J+- in £ and 77 direction here will be treated generally for variable q overall in the flow field as
where/and g are convection terms in £ and 77 direction, U and V are defined in eq. [3] and [4]. Based on our experience the numerical constants r, £, C, and C2 can be selected as r=1.5, e=10-10, Cy=2.0, C2=1.5 - 8.0. This formulation is defined to be scheme which satisfies the condition that in the limit as the mesh width Ax —> 0, local maxima are non-increasing, and local minima are non-decreasing.
2.2.2 Moving grids technique To simulate the flow field containing electrical contacts moving during the current separation, two types of moving grids are generally used. The type I moving grid moves only its form and the number of grid will be not changed as shown in Fig. la. In contrast with the type I, the type II moving grid will be expanded or compressed by adding new grid lines to or removing existing ones from left or/and right side as shown in Fig. 1b. Some methods have to introduced to treat both types of moving grids correctly.
Figure 1. Two types of moving grids
Fields of application
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2.2.2.1 Treatment of moving grid type I In the type I moving grid, the variables will be kept in the same cells during the movement at a time step. This will produce error if no additional measures will be taken, because the positions of these values in the flow field have been changed through the moving of the grid cells. To avoid such errors, additional terms in eq. [1] must be introduced. In the case that a block of grid moves with a speed of XT and yT as its x and y components, the velocities U and V in the curvilinear coordinate system with £ = %(x,y,t) and 77 = Tf(x,y,t) will then be (s. [STE 78]):
where <^T = -x^x -y£y and %r = -x£x -y£y, then the convection term in eq. [1] after the transformation to the curvilinear coordinate system is
2.2.2.2 Treatment of moving grid type II For the type II moving grid, the movement only involves the two side grid lines, i.e., almost all the cells of this grid are not in motion, therefore, there is no need to apply the additional terms. However, because of the change in the cell size at every time step, there would be implicit unphysical energy and mass loss as grid compresses and unphysical energy and mass increase as the grid expands, if the values of energy and mass after the change of cell size were taken simply from the old before the change of cell size. Therefore, sophisticated treatment of the value of energy and mass is very important to get the accuracy of calculation. As grid expands or compresses, the value of the density and the total energy in the changing cell after the change of cell size will be then
where Xj.j-Xi is the cell size, new and old denote the values after and before the change of cell size.
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Finite volumes for complex applications
2.2.3 Chimera boundary Because of the movement relative to neighbouring grids, the moving grids will have discontinuity of grid line at interfaces with their neighbouring grids. Hence, the Chimera interpolation method (s. [STE 87]) will be applied to enable the information exchange between two grids with discontinuity of grid lines. The basic concept is that two layers of ghost cell of a block will overlap with the neighbouring block. To obtain values in the ghost cell, the values of the nearest cell centres of the neighbouring block are interpolated to the cell centre of the ghost cell (s. Fig. 2). The values in the ghost cells can then be used for flux building. Same as the central discretisation, the Chimera interpolation is non-conservative, however, the upwind character is obtained through the ELED scheme.
Figure 2. Overlapping blocks for Chimera interpolation 3. Results and Discussion 3.1 Improvement of numerical scheme To investigate the improvement of the numerical scheme especially in its capability to resolve the flow discontinuity and to keep a small disturb in shear layer, two test cases are considered.
Figure 3. Two test cases for numerical schemes
Fields of application
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The first test case is the shock, tube problem as described in [SCH 98]. In Fig. 3a, the pressure distributions calculated with the JST and ELED schemes are compared with the exact solution of [HIR 90] at the time t=6.1 ms. As can be seen, the scheme of JST oscillates strongly just before the shock front whereas the ELED scheme resolves the discontinuity without oscillation. The use of ELED scheme resulted in only 7 percent increase in the computing time. The other test case is the fully developed turbulent pipe flow as shown in Fig. 3b, where the calculated fully developed radial velocity distributions, which is normalised by the velocity in the middle of pipe Um, are compared with the experiment results of [NIK 32]. It can be seen that the near-wall shear layer is strongly disturbed by the JST scheme, while the result of ELED scheme agrees very well with the experiment.
3.2 Verification with experiments of a circuit breaker Fig. 4 shows a schematic diagram of the core part of a circuit breaker. To interrupt the current, the electrical contact "finger" will be moved together with isolating and auxiliary nozzles toward left, while the electrical contact "plug" will stay still. The gas in the pressure chamber (not illustrated) will be then compressed through this motion and flows from left into the isolating nozzle, there will be a highly transient and transonic flow.
Figure 4. Schematic diagram of the simulated circuit breaker To verify the code for the capacitive switching design, measurement of static pressure on the 6 points showed in the Fig. 4 was carried out for cold. These 6 measured points are located in different flow regions. ® and (D are in the diffuser region with flow separations; ® is in the geometric throat; ® and © are in the channel flow region with boundary layer character; (D is behind the shock front and presents the pressure lost over the shock. Therefore, the values at these 6 positions reflect all flow details. For the calculation, an inlet was defined as shown in Fig. 4 and the pressure measured in the pressure chamber was used as the inlet boundary condition, so that the leakage in the pressure chamber which is difficultly to be estimated can be ignored. In Fig. 5 the simulated pressure distributions at © to ® are
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Finite volumes for complex applications
Figure 5. Comparison of pressure in circuit breaker: simulation, O measurement
Fields of application
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compared with the measurement. The simulation results agree very well with the measurements. The discrepancy of P5 is caused by the selection of the inlet where the boundary layer begins downwards from the start point in the reality.
3.3 Coupling of the electrical and flow fields Based on the test results on a circuit breaker where breakdown voltages were measured in dependence of travel positions, the present code can be verified through comparison of the ratio E/(pkr) with the critical value of 1480 kVm 2 /kg for 6 bar absolute filling pressure of SF6 from the streamer theory (s. [BEY 86]). In this case E/(pkr) must be higher than the critical value, where kr is a roughness factor which accounts for the microscopic effects which intensify the local field strength and is obtained from an semi-empirical function of local density and roughness. The electrical field strength E was calculated with the ABB electrical field program ACE. The gas density p was obtained from the flow simulation with the present code. Fig. 6 shows the distribution of E/(pkr) in the circuit breaker. The maximum of this ratio lies on the surface of the plug contact and it overruns the critical value slightly. The simulation results correspond very well with measured breakdown/hold values of the applied voltage observed in measurements.
Figure 6. E/(pkr) distribution and its maximum at travel=l 15 mm
4. Conclusion The progresses in the numerical methods, including numerical scheme and moving grid technique, have been made, leading to the development of the code presented in this paper which is able to fulfil the requirements of capacitive design of high voltage circuit breaker. The effectiveness of the code is confirmed through the following facts: (1) The adoption of the more advanced ELED scheme resolves flow discontinuity efficiently; (2) the physical viscosity is not confused; (3) the calculation results agree very well with the measured results; (4) the calculated
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density was coupled with electrical field strength, and the resulted ration of E/(p kr) predicted the gas breakdown correctly. The code will be further improved and developed, in particular, arcing model will be implemented and verified, so that it can be used to predict the dielectric strength under high temperature and pressure.
5. References [BEY 86] Beyer, M, Boeck, W., Moller, K., Zaengl, W.: Hochspannungstechnik, SpringerVerlag, Berlin Heidelberg New York, 1986 [CLA 97] Claessens. M., Moller, K., Thiel, H.G.: A computational fluid dynamics simulation of high- and low-current arcs in self-blast circuit breakers, J. Phys. D: Appl. Phys. 30, p. 2899-2907, 1997 [HIR 90]
Hirsch, C.: Numerical computation of internal and external flows, Vol. 2, John Wiley & Sons, 1990
[JAM 81] Jameson, A., Schmidt, W., Turkel, E.: Numerical solutions of the Euler equations by finite volume methods with Runge-Kutta time stepping schemes, AIAA paper 81-1259, January, 1981 [JAM 94] Jameson, A.: Analysis and design of numerical schemes for gas dynamics 1: Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence, Int. J. of Computational Fluid Dynamics, August, 1994 [LAU 74] Launder, B.E., Spalding, D.B.: The numerical computation of turbulent flows, Computer Methods in Applied Mechanics and Engineering, Vol. 3, p. 269-289, 1974 [NIK 32]
Nikuradse, J.: Gesetzmafiigkeit der turbulenten Stromung in glatten Rohren. Forsch. Arb. Ing.-Wes. Heft 356, 1932
[SCH91] Schafer, O.: Application of a Navier Stokes Analysis to turbomachinery bladecascade flows, 19th International Congress on Combustion Engines, CIMAC, Florence, 1991 [SCH 98] Schafer, O. et al: Last advances in numerical simulation of aerodynamic forces on turbine blades of turbochargers for pulse charged engines, 22nd CIMAC , International Congress on Combustion Engines, 19-21 May , Kopenhagen , 1998 [STE 78] Steger, J.: Implicit finite-Difference simulation of flow about arbitrary twodimensional geometries, AIAA Journal, Vol. 16, No. 7, July, 1978 [STE 87]
Steger, J., Benek, J.: On the use of composite grid scheme in computational aerodynamics, Computational Methods in Applied Mechanics and Engineering, Vol. 64, No. 1-3, 1987
[TRE 91] Trepanier, J.Y. et al: Analysis of the dielectric strength of an SF6 circuit breaker, IEEE Transaction on Power Delivery, Vol. 6, No. 2, April, 1991
River valley flooding simulation
Francisco Alcrudo Area de Mecdnica de Fluidos Maria de Luna, 3 CPS - Universidad de Zaragoza 50015 Zaragoza, SPAIN
ABSTRACT
Dam break flood wave propagation along a reach of a river valley located in the Italian Alps is mathematically modeled with package SW2D that solves the nonlinear Shallow Water equations. Simulation results are compared with data obtained from a physical model of the river valley operated by ENEL (Italy). The difficulties encountered during the modelisation process and the solutions adopted are explained in this paper. Key Words: Dam break, Flood, Shallow Water, Physical Model
1. Introduction Considerable efforts are being presently devoted to the validation of numerical models describing dam break flows, mainly due to the need for modern risk assesment and mitigation tools. Real life experimental data concerning actual dam break or severe flooding are very difficult to obtain because of the unpredictable nature of the phenomenon. However, measurements obtained from reduced scale physical models can provide excellent validation information because the experimental conditions can be more precisely defined. The work reported in this paper concerns the comparison of the simulation results obtained with SW2D program [ALC92] with measurements of the flooding experiments carried out by ENEL (Italian Utility Company) in a reduced scale physical model of the Toce river. The physical model is some 50m long by llm wide and is built mainly in concrete (see figure 1). It represents a 5km long reach of Toce river which is located in the Italian Alps.
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Figure 1: Digital Terrain image of Toce river valley physical model (ENEL) The model reproduces many details of the actual valley geometry including the river bed and some villages, hydraulic structures and a reservoir located in the middle of the reach that depending on the intensity of the flood is overtopped and eventually filled with water across its embankments. The upstream end of the river reach model is connected to a small water tank fed by a hydraulic pump. Flooding is initiated by starting the pump that rapidly fills up the tank, overtops the entrance to the reach and rushes downstream. The pump capacity is such that the process takes place very rapidly thus simulating an abrupt irruption of water into the valley model. ENEL personnel located water stage probes at 32 different positions in the model valley. Among them one in the river bed at the entrance section and another one in the middle of the feeding tank that can be used to impose the boundary conditions together with the pump discharge versus time that was also recorded. Experiments were carried out for two flooding intensities: The first one such that no overtopping of the reservoir takes place (peak discharge of 0.21 m 3 /s) and the second one with reservoir overtopping (peak discharge of 0.36m 3 /s). Measured water stage readings at several probe locations were compared to water levels obtained with SW2D model for different friction coefficients and flood intensities and overall satisfactory agreement was found. The physical model geometry was distributed by ENEL as a Digital Terrain Model (DTM) covering the model area at regular intervals of 5cm, therefore specifying the bottom elevation function z s ( x ^ y } in some two hundred thousand points. The simulations reported here were run on a platform of comparable computing power to that of a Pentium processor. In order to have reasonable run times (a few hours) the size of the DTM grid had to be coarsened by a factor of three.
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2. Mathematical model It is commonly accepted that bulk flood flow can reasonably be well described by the non linear Shallow Water equations that simply express conservation of mass and momentum in the plane of movement of water. Since they are obtained after and integral mass and momentum balance in the horizontal directions (or by averaging the Navier-Stokes equations across the vertical) no information regarding vertical velocities is obtained. Usually no flow shear forces are taken into account when the problem is convection dominated as it is the case in severe flooding. Friction forces with the bottom are accounted for by empirical formulae such as Manning's or Chezy's. The Shallow Water equations can be written in integral conservation vector form as follows:
Here t represents time, dV an elementary volume and nx and ny the cartesian components of the normal vector to the elementary surface area dS enclosing the considered volume. Think that in 2-D a volume means in fact an area and an area is actually a line. U is the vector of conserved variables and F and G are the cartesian vector fluxes of mass and momentum.
Here h, u and v represent water depth and the two cartesian velocity components respectively and g is the acceleration of gravity. Source term H accounts for bed friction and bottom slope:
where n is Manning's friction coefficient. The mathematical model SW2D solves the Shallow Water equations in two dimensions by means of a finite volume spatial discretization in multiblock structured meshes coupled to an explicit two step time integration scheme. This is done by applying equation (1) to every cell of the computational domain in the usual Finite Volume approach. Cells can be quadrangles of arbitrary shape but sound judgement has to be exercised so as to avoid very distorted or stretched control volumes that may degrade the overall accuracy. Numerical fluxes are evaluated at cell faces through MUSCL variable extrapolation with limiting to enforce monotonicity. After variable extrapolation, Roe Riemann solver is applied at each cell interface. Bottom slope and bed friction represented by Manning's formula are spatially integrated pointwise.
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Figure 2: Cartesian grid used in the computations totalling 22000 points Once the spatial discretisation has been done the solution is advanced in time by a predictor-corrector sequence. Source terms are implicitly time integrated with no extra cost because the operator remains diagonal if they are pointwise spatially discretised. Details of the algorithm can be better found in [ALC98]. 3. Boundary conditions Proper computation of the flow variables at the upstream end in order to reproduce the correct flood characteristics at the inflow of the model valley is crucial if good agreement with downstream located probes is sought. Downstream boundary conditions do not exert such a strong influence on the global flowfield mainly because water leaves the reach in critical or supercritical conditions. Available initial data from ENEL were the inflow rate, Q, the reading of the water level probe located in the inflow tank, named SI, and the reading of the water level probe located at the inlet section, named S2. After considerable efforts it was determined that good agreement with experimental data at the entrance could be found only by imposing a subcritical inflow condition based on the available total head at the feeding tank. Despite the advise given by ENEL that flow conditions are critical at the entrance section, the model could not be run under this assumption. Numerical experiments showed that failure to accept critical flow at inlet was due to a slight adverse slope in this area that led to flow reversals, because critical flow can only be reached at the top of an upslope. Since subcritical inlet conditions require that two flow variables be imposed, they can be implemented by either imposing flowrate, Q, water level at the inflow section (S2 probe reading) or far upstream on the reservoir (SI probe reading) together with the inlet angle, a.
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Imposing flowrate is not adequate in a two dimensional computation involving an irregular inlet section because it is difficult to obtain an appropriate criterion to distribute the available discharge among the inlet section cells. Using the water level at the inflow section is an interesting option but it is better to use the water level at the feeding tank (SI probe reading) as the Total Head available in order to have water level at the inlet section (S2 probe) as an accuracy check. It must be borne in mind that the inflow rate, Q, can also be used as a check. Due to the size of the tank with respect to the entrance to the reach, the velocity in the former can be considered negligible, and the reading given by probe SI is considered as the total head /IT that is available at the inflow section. This can be written as follows:
being (h)si the reading of probe SI at the considered time and the other variables with subindex inflow are evaluated at every cell of the inlet section. Also from the outgoing bicharacteristic (see [ALC92] one has: (11-11 + 2c)inflow = (u • n + 2c)Mcfc
(5)
where u is the velocity vector and n the locally outward pointing unit vector. Subindex bich corresponds to the expression transported by the outgoing bicharacteristic from the inside of the computational domain. Once water depth and modulus of the water velocity at every inlet section cell are determined from the above equations the two cartesian components can be computed if an inflow angle, a, is imposed. In the tests run a was varied from zero to a few degrees with no significant changes in the computed results. 4. Testcases Besides the two inflow hydrographs (of different intensities) tested by ENEL, several simulations were performed varying Manning's friction coefficient above and below the value of n=0.016 suggested by ENEL. Also and more importantly, runs were made both with and without the buildings composing the valley villages. The DTM geometry did not contain buildings, but these could be included by modifying the bottom surface function Z B ( x , y ) appropriately. However, due to the low resolution of the grid used, buildings are represented very roughly as figure 1 shows: Villages can be seen as groups of mushroom like sprouts. Their influence on the solution is nevertheless very substantial. For every run made, great attention was paid to matching the inflow rate supplied by the pump (Q) and the water level measured by the probe located in the river bed at the inlet section (S2 probe). This guaranteed that at least the inflow flood wave was close to the actual one. Figures 3 and 4 show the
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Figure 3: Inflow rate and S2 probe reading for the moderate flood event
Figure 4: Inflow rate and S2 probe reading for the severe flood event comparison between computed and measured inflow discharge (left) and water depths in centimeters (right) at probe S2 (located just at the inflow section) versus time for the two tested flood events. Computational results (crosses, plusses and circles) correspond to different model options as shown. Although agreement in both flow and water depth is quite good, runs with buildings follow better the experimental values. Figures 5 and 6 show the comparison between calculated and measured water levels at probes S4 and P8 for both flooding events. Both probes are located around the central part of the valley, S4 being about 5m and P8 some 16m downstream of the inlet section. Although runs without buildings show larger errors than those including buildings (especially at probe S4) the situation is reversed at other probes not shown here for lack of space. Finally figures 7 and 8 show the same comparison at probes named P13 and P21. Both lie close to the river bed. P13 is located in front of the central reservoir about 21m downstream of the inlet section while P21 is located some 7m further downstream. Overall agreement at the considered locations can be judged acceptable.
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Figure 5: Comparison at S4 and P8 probes for moderate flood
Figure 6: Comparison at S4 and P8 probes for severe flood
Figure 7: Comparison at PI3 and P21 probes for moderate flood
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Figure 8: Comparison at P13 and P21 probes for severe flood 5. Concluding remarks Although discrepancies between measured and computed water levels can be important at certain probe locations, the mathematical model used provides a reasonably accurate description of the two flooding events considered. Due to the very valuable assistance that this kind of tool can provide in tasks such as land use and emergency planning or risk assesment studies, it seems worthwhile to carry out further validation and improvement work. 6. Acknowledgements The author would like to thank ENEL and especially Dr. G. Testa for providing the experimental data and clarifying many technical questions. Finantial support provided by the European Union under CADAM concerted action is also gratefully acknowledged. 7. References [ALC 92]
[NUJ 95]
[ALC 98]
ALCRUDO F., Esquemas de alta resolution para el estudio de flujos discontinues de superficie libre, Ph.D. Thesis, Universidad de Zaragoza, 1992 NUJIC M., Efficient Implementation of non-oscillatory schemes for the computation of free surface flows, Journal of Hydraulic Research, 33, No. 1, 1995, p. 101-111 ALCRUDO F., Dambreak flood simulation with structured grid algorithms , Proceedings of the 1st CADAM (Concerted Action on Dam Break Modelling) Meeting, (1998), Published by the EU, in press.
Modelling vehicular traffic flow on networks using macroscopic models J.P. Lebacque1, M.M. Khoshyaran2
1
CERMICS-ENPC. TASC. USA.
2
FRANCE, email: [email protected].
ABSTRACT: In this paper, we describe a macroscopic model for vehicular traffic flow, with several extensions, resulting in a flow model on a network. These extensions require the introduction of link boundary conditions, partial flow dynamics and intersection models. Some numerical schemes based on the Godunov scheme are proposed for the discretization of the model. Key Words: Godunov scheme, Traffic flow, LWR model, partial flows
1
Introduction
Macroscopic modelling of vehicular traffic flow goes back to the pioneering work of Lighthill and Whitham [LW 55] and Richards [Ri 56], which introduced the celebrated LWR (Lighthill Whitham Richards] model of traffic on an infinite track. This model relies on the continuum hypothesis, i.e. the asumption that vehicular traffic can be described by macroscopic variables, the density K(x, t), the flow Q(x,t), and the speed V(#,£), as functions of the position x and the time t. These variables are related by the following equations:
or simpler:
Qe and Ve represent the equilibrium flow-density resp. speed-density relationdef
ships ( Q e ( K , x ) = KVe(K,x}). Their aspect is the following:
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Of course, considering the continuum hypothesis, the above LWR model (1) should be considered as a phenomenological model, but it is usually accepted that it provides a reasonably good description of the dynamics of traffic flow at a space scale of a hundred meters and a time scale of 10 seconds. Actually, the LWR model (1), also refered to as the first order macroscopic traffic flow model, constitutes but one among several competing approaches to macroscopic traffic flow modelling (see [LL 99] for a general discussion). Other notable modelling approaches include the second order macroscopic traffic flow models (see [Le 95] for an overview and [Sc 88] for relations between second and first order models) and the kinetic traffic flow models [PH 71], [Ph 79], [He 97]. Neither experimentation nor theory has provided arguments strong enough to support one model unambiguously. There is also no real consensus concerning the exact functional form of the equilibrium relationships, but the shapes suggested in the above illustration are generally accepted, up to a few variations. In the sequel, we shall concentrate on the LWR model, which is simple, enjoys obvious physical meaning, and provides results generally in good agreement with measurements.
2
The Godunov scheme for the classical LWR model on the line
The entropy solution of (1) is the only solution considered usually in the literature on traffic flow modelling. In entropy solutions, the decelaration of trafic generates Shockwaves, whereas the acceleration of trafic induces rarefaction waves. Entropy solutions are also characterized by the fundamental fact that they maximize locally the flow [Le 96]. The Godunov scheme [GR 91], [Kr 97] provides a numerical solution of the classical LWR model, as shown in [Le 96], [Da 95]. This solution is satisfactory for applications: it approximates the entropy solution. Let us introduce the equilibrium supply and demand functions:
(the symbols + and - represent right- and left-hand limits). The following illustration describes these functions, that represent respectively the greatest possible inflow (supply) and the greatest possible outflow (demand) at point x.
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With these notations, the expression of the Godunov scheme is straightforward:
with as usual (i) = [%i-\, Xi] the cell i of length li, t the index of the time-step, Kti the average density in cell i at time t At, Qti the average flow at point Xi during time-step t. The flow equation expresses that the flow is the minimum between the downstream supply and the upstream demand. The flow equation in (4) is the expression of the analytical solution of the Riemann problem, which can be obtained even if the upstream and downstream equilibrium relationships differ [Le 96]. The flow maximizing property of the entropy solution is crucial for this result. The practical necessity for considering space-dependent and even space-discontinuous equilibrium relationships is obvious: lane drops, intersections provide contexts for such discontinuities.
3 3.1
Extensions of the basic LWR model Link boundary conditions
For applications, extensions of this basic LWR model are indispensable. A first and obvious generalization concerns the extension of the model to networks, which implies two steps: the definition of proper boundary conditions for links, and the description of intersections. The equilibrium supply and demand concepts provide the proper framework for the definition of link boundary conditions. Considering now a link such as the following:
the boundary conditions are the upstream demand A u (£) and the downstream supply Sd(t) [Le 96], [LK 98]. The link inflow Q(a,t) at any time is the minimum between the link supply S e (K(a+, t ) , a ) and the upstream demand A u (t). Similarly, the link outflow Q(b, t) at any time is the minimum between the link demand Ae (K(b—,t),b) and the downstream supply £d(t). Thus:
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Partial flows on links
Before considering intersections, it is necessary to consider partial flows on links. Indeed, the fashion in which the traffic flow separates (according to preselection lanes) or does not separate (so-called FIFO flow) in the incoming links of an intersection determines the way in which the intersection works. Further, in many advanced applications, various categories of users must be considered: users would be distinguished according to destination, information availability, path, etc. The macroscopic variables are disaggregated according to some assignment attribute d:
The partial flows and densities are related by the trivial relationships:
These equations must be completed by a phenomenological model. The simplest possible model is the so-called FIFO model:
(vehicle speed independent of attribute d). This model results in a straightforward advection equation relative to the composition coefficients Xd — Kd/K of the flow: in which the global velocity of the flow V results from the resolution of (2)). A more realistic model is the lane assignment model [LK 98]. In this model, vehicles may have restricted access to lanes according to the assignment attribute d. Let / be the set of lanes, Id the set of lanes accessible to vehicles d, JiKmax the maximum density of lanes i, Kf the density of vehicles d in lanes i. Then the Kf are the unknowns of the lane-assignment problem and are subjet to the following constraints:
The Kd constraints express the split of Kd into the Kf, and the1/2Kmax constraints express that the total density in lane i cannot exceed the maximum density riK max of this lane. The Kd constitute the dynamic data and the
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7i and Id constitute the geometric data of the lane assignment problem. The unknowns K? can be determined by solving either
(maximizing locally the total flow), or
(Wardrop optimum), subject to constraints (9) in both instances. The meaning of (11) is to assign users to lanes in such a way that all users having the same attribute d have the same speed on all the lanes they use effectively (otherwise, users would switch lanes in order to drive faster: this is an individual optimum). These lane assignment models result in systems of conservation laws for which approximate Riemann solvers are under study. A simple case (2 user types, 2 lanes) was analyzed in [Da 97]. Supply-Demand models for partial flows can also be defined. The principle is to calculate partial supplies and demands for all superscripts d and to determine the corresponding partial flows by comparing partial demands to partial flows. Partial demands are defined as:
which is a FIFO-like model. The partial supply model really defines the user behavioral model. Let us first define coefficients fid which determine the maximum density (i.e. fidKmax) of the lanes available to vehicles d. If we refer to the notations of the preceding subsection,
We propose the following two models for the partial supplies: Model 1: £ d (z,f) = (3dY,(x,t) (linear model), Model 2: E d (z,£) = (3dZe ( K < 1 ( x + , t ] ,x) (homogeneous section model). Model 1 is extremely simple but allows Kd to exceed PdKmax. Model 2 does not have this drawback but still does not take partial flow overlapping into account as precisely as the lane-assignment models, since the data formed by the Id sets and the coefficients 7$ has been simplified and only coefficients (3d are left. It would also be natural to define the partial flows Qd as: Nevertheless, since ^j3d is usually > 1 (because of the partial flow overlapd
ping), it is possible that partial flows calculated according to (13) satisfy:
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thus implying a model inconsistency. This inconsistency can be resolved by using the following expressions:
The partial flows Qd can be viewed as solutions of the following program:
with (pd some concave increasing functions such as: I he above functions
3.3
Modelling intersections
Modelling intersections is more difficult, since intersection models are phenomenological by nature. They describe for instance, in the case of a merge, the local priority rules, or the gap acceptance process. Two modelling schemes can be considered. 1. Modelling intersection as objects of finite extension, by trying to reproduce the movement dynamics. This was the idea of the STRADA model [BLLM 96], in which exchange zones generalize cells inasmuch as they behave similarly but are endowed with several entry- and exit- points and provide upstream demands respectively downstream supplies for downstream respectively upstream cells or exchange zones. Exchange zone models are discrete by essence. 2. Pointwise intersection models. These were considered in [Le 96] [LK 98] and derived from zone models by letting the zone extension become vanishingly small. The study of these models is the subject of ongoing research. Let us still give one example and consider the node depicted hereafter. Let us denote Ej(t) the supply of exit link (j) of the intersection at the node point, and Aj(t) the demand of the entry link (i) of the intersection at the node point. A proportion 7ij of users about to exit link (i) chooses link (j) (the coefficients jij are called assignment coefficients and must be considered exogeneous to the flow model). Thus the partial demand of traffic from link (i) to link (j) def is given by: Ay-(t) = 7jjA;(£). Split supply coefficients bij (depending on the link geometry) can be introduced in order to disaggregate the link supplies £j(£). We can deduce partial supplies Sij(i) by applying for instance model def I (the simplest): %ij(t) — bijEj(t). Since it is possible and even likely that
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1, a formula similar to (14) should apply, with the same rationale, to yield the partial flows
4
Discretization of partial flows in links and intersections
Let us consider the Supply-Demand model for partial flows (notably equations (12) and (14). If we consider two consecutive cells (i}, (i + 1), the following relationships result (discretized model): (expressing the traffic supply of cell (i + 1) as a function of the cell mean density K\+l, (expressing the partial supplies, according to partial supply models 1 or 2, (expressing the traffic demand of cell (i) as a function of the cell mean density expressing the partial demands according to the FIFO-like model, expressing the partial flows between cells (i) and (i + 1) according to (14), expressing the discretized conservation equation, yielding the total cell density and flow as the sums of partial densities and flows. The similitude between the intersection model (16) and the Supply-Demand partial flow model (12), (14) is evident, thus the discretization of the intersection model follows the same lines as the discretized Supply-Demand partial flow model and need not be described in detail here.
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5
Conclusion
The development of extensions of the basic LWR is still an ongoing process. The only intersection model for which there exists any kind of experimental support is the Supply-Demand model [LK 98], and link partial flow models are still tentative. Nevertheless, suitable discretized models should be developed, in order to be able to choose between alternative modelling schemes. The finite volume method, combined with the search for analytical solutions, seems to be the best approach to the investigation of numerical solutions of the LWR model and its extensions.
References [BLLM 96] C. Buisson, J.P. Lebacque, J.B. Lesort, H. Mongeot. The STRADA model for dynamic assignment. Proc. of the 1996 ITS Conference. Orlando, USA. [Da 95] C.F. Daganzo. A finite difference aproximation of the kinematic wave model. Transportation Research 29B. 261-276. 1995. [Da 97] C.F. Daganzo. A continuum theory of traffic dynamics for freeways with special lanes. Transportation Research 31 B. 83-102. 1997. [GR 91] E. Godlewski, P.A. Raviart. Hyperbolic systems of conservation laws. SMAI. Ellipses (Paris). 1991. [Kr 97] D. Kroner. Numerical schemes for conservation laws. Wiley Teubner. 1997. [He 97] D. Helbing. Verkehrsdynamik. Springer Verlag. 1997. [Le 95] J.P. Lebacque. L'echelle des modeles de trafic: du microscopique au macroscopique. Annales des Fonts. 1st trim., 74: 48-68. 1995. [Le 96] J.P. Lebacque. The Godunov scheme and what it means for first order traffic flow models. Proc. of the 1996 ISTTT (J.B. Lesort ed.). 647-677. 1996. [LK 98] J.P. Lebacque, M.M. Khoshyaran. First order macroscopic traffic flow models for networks in the context of dynamic assignment. EURO Work Group on Transportation 1998, Goteborg (Sweden). CERMICS Report. To be Published. [LL 99] J.P. Lebacque, J.B. Lesort. Macroscopic traffic flow models : a question of order. 14th ISTTT. Accepted for publication. 1999. [LW 55] M.H. Lighthill, G.B. Whitham. On kinematic waves II: A theory of traffic flow on long crowded roads. Proc. Royal Soc. (Lond.) A 229: 317-345. 1955. [Ph 79] W. F. Phillips. A kinetic model for traffic flow with continuum implications. Transportation Planning and Technology, Vol. 5, 3, pp 131-138. 1979. [PH 71] I. Prigogine and R. Herman. Kinetic theory of vehicular traffic. American Elsevier, New York. 1971. [Ri 56] P.I. Richards. Shock-waves on the highway. Op. Res. 4: 42-51. 1956. [Sc 88] S. Schochet. The instant response limit in Witham's non linear traffic model: uniform well-posedness and global existence. Asymptotic Analysis. 1, pp 263-282. 1988.
Finite volume method applied to a solid/liquid phase change problem
El Ganaoui M., Bontoux P. IRPHE-Umversite d'Aix-Marseille II IMT, 38 Joliot Curie 13451, Marseille Mazhorova O. Keldysh institute Moscow
ABSTRACT A second order accuracy method of time and space based on finite volume approximation in a fixed m.esh is developped for Navier-Stokes and energy equations extended to solid/liquid phase change problems. This fixed grid method validated with respect to an interface tracking method is able to describe the interaction of steady and oscillatory melts with the interface during Bridgman crystal growth. Key Words: Finite volume, phase change, interface.
1. Introduction Free and moving boundary problem requires the simultaneous solution of unknown field variable and the boundaries of the domains on which these variables are defined. Phase change during directional solidification of semiconductor crystals by the Bridgman technique is a typical example of such a complex process. Each method of solution must solve the appropriate heat, mass and momentum transfer equations and determines the melt solid inter-
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face. It was still necessary to satisfy the Stefan or similar derivative condition on that boundary. Furthermore it was sometimes be difficult or even impossible to track the moving boundary directly [CRA 84]. The possibility, therefore, of reformulating the problem in such a way that the transmission conditions at the interface are implicitly bound up in a new form of the equation, which applies over the whole of a fixed domain. The moving boundary appears, a posteriori as one feature of the solution. One possibility of reformulating the problem is to introduce an enthalpy function in the energy equation and a porous model in the momentum equation. The enthalpy function is the sum of the specific heat and the latent heat required for the phase change. In the momentum equation, we assume that the liquid turn to solid in an intermediate region to be a porous medium. In this way on prescribing a Darcy source term the velocity value arising from the solution of the momentum equation are inhibited, reaching values close to zero on complete solid formation [VOL 80]. The coupled enthalpy porosity model gives a single set of homogenous NavierStokes and energy equations adapted to the problem of phase change during directional solidification [MOR 99]. The finite volume method is validated with respect to an interface tracking method [ELG 96]. It uses a fixed grid and the interface position is given from the thermal field (solidification isotherm). The resulting interface shape is also studied and some insight on cristal constitution are given. 2. Formulation
For directional solidification, A cylindrical ampoule with radius R and length L contains melt and crystal. The ampoule must be moved relative to a prescribed external temperature gradient. This motion of the ampoule is acounted for by supplying a melt to the top of the computational space at a uniform velocity Ut and with drawing cristal from the bottom with the same velocity. The heat transport between the furnace and the ampoule is modelled with a prescribed furnace temperature profile with three zones, cold (T = T c ) , adiabatic (dT/dn = 0) and hot one (T = T/j). The length L, the velocity a/L and the thermal difference Th — Tc are used as reference scales to give dimensionless form of the variables ( x , u , 6 ) . x ( r , z) represents the courant point with radial r and the axial z components, u = (ur,uz] represents velocity with radial and axial coordinates ur and uz, respectively. Liquid, solid and intermediate medium are distinguished by the suffixes /, s and si. For the energy equation a continuous enthalpy function is introduced :
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where £ is a prescribed small regularisation of the temperature, /; a mesure of liquid fraction and Ste = Lj/c(Th — Tc) is the Stefan number. The corresponding enthalpy is continuous and piecewise linear. In this way the energy equation takes the following form in all the dimensionless domain 0 < r < 1 et Q
If Ste 1 goes to 0, the equation (2) goes to the classical one phase energy equation. For the momentum equations a permeability term is introduced:
The momentum equations takes the following form :
where Ra = gfl(Th — T c )/i/a, is the Rayleigh number, Pr — i//a is the prandtl number characteristic of the material and Da = L 2 /A'o is the Darcy number characteristic of the phase change morphology. This equation contains asymptotically the behaviour of the velocity in each media. When A' = A"/, the term K(x,t) lu tends to zero and in the fluid domain QI and we solve a near approximation of the Navier Stokes-equations. When K = K s , the term K(x,t)~1u penalises the momentum equation, the other terms of the equation become negligible and implie us ~ 0. Equations (2) and (4) must be added to the continuity equation:
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Finite volumes for complex applications
3. Solution method
3.1 Finite volume method Equations (2), (4) and (5) are discretized by the finite volume method [EYM 97]. To explain the approach we consider the two-dimensional convection diffusion equation for a general variable (p with the velocity field in cartesian coordinates u :
f ( ( f ) = u/f — 7^3 grad
where TV(X) is the normal vector to the boundary dV at point x, outward to V. In order to define a finite volume scheme, the time derivative is approximated by a finite difference scheme on an increasing sequence on time ( t n ) n ^ \ N with *0 = 0. A mesh M of the physical dom ain ft of IR2 (£7 = UV, V 6 M] is introduced. The discrete unknowns at time tn = nSt (St is the time step), are expected to be an approximation of (p on the cell V around the point Mij and noted 0"^. Equation (6) is integrated over each cell V using the Gauss divergence theorem :
where (d(p/dt)n is given by the time scheme at the time step tn = n St in the control volume V. The next step is the approximation of the convective part Fc .TV and the diffusive part Td.TV of the projected flux T .TV over the boundary dV of each control volume.
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3.2. Time discretization The time integration is performed implicitly using a three levels, given a second order truncation error in time.
3.3 Flux approximations The diffusive part of the flux is discretized with a second order truncation error in space:
Different discretizations for the convection are possibles, central schemes apply a symmetric interpolation for ^'+1/2 > Upwind schemes apply a one side interpolation. Leonard [LEO 90] has introduced a QUICK and other scheme as a mixtures between the two kinds of interpolation. We have used QUICK scheme with a second order truncature in space. we write f f + 1 / 2 t j = ui+i/2,j
the subscript m is equal — 1 for w;+i/2,j > 0 and 0 for Uj+i/2,j < 0. 4. Results
We consider a two-dimensional model in cartesian coordinates corresponding to a diameter section of the growth ampoule. This model permits the investigation of asymmetric flow. We consider unstable configuration (IVB), which foster the amplification of hydrodynamic instabilities. In this case the computation exhibits steady symmetrical solution up to Ra = 8 x 104. This solution is characterized by the presence of two contra-rotating cells with equal sizes and a flat interface. Transition to asymmetrical solutions occurs
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Finite volumes for complex applications
at Ra = 1.6 x 105. The asymmetrical instability is shown to become timedependent at 4.2 x 105 < Ra < 6.4 x 105 and interacts with the interface shape. A periodic solution is obtained for Ra — 8 x 105. A description of the unsteady regime effect on the interface shape during the growth process is given in figure 1. In a space of time corresponding to one period of the periodic regime, the two contra-rotating cells which compose the melt exchange their sizes in time: at t = to + T/6 (T is the period) the flow is dominated by the cell of the right which brings about the interface, with the presence of a second cell in the left. This situation changes at t = to + T/3 in favour of the left cell which becomes dominated at t = t0 + T/2 and configuration is still the same until t = to + 5T/6. The interface interacts with the previous hydrodynamic regime. The crystal characterised by successive solidification interfaces is represented in the figure 2 on the diameter crystal. This result shows a significant structure which appeared to be relevant of some patterns exhibited in the experiment. 5. Conclusion
The finite volume method is able to simulate accurately complex physical phase change problem. It represents the flow pattern and the heat transfer during directional solidification. The method enables to determine the various symmetric, asymmetric, steady and time-dependent flow solutions and to analyze the transitions between regimes. A recent extension concerns the effects of inequal thermal conductivity in solid and liquid phases on the interface shape and hydrodynamic transitions. A cknowledgment
We would like to thank Dr J. Ouazzani from Arcofluid and Dr D. Morvan from IRPHE for helpful and comments. We thank also the CNRS for support on CRAY C98 of IDRIS-CNRS Center in Orsay. Support from CNES are also acknowledged.
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Figure 1: Description of various hydrodynamic configurations during one period of the periodic regime Ra = 8 x 105. The signal of the temperature T"(io,io)(i) is represented in the point (10,10) in the melt function of time (t — tinit)/6t,6t = 10~ 4 .
Figure 2: Resulting crystal by considering successive interfaces during one period of the periodic regime Ra = 8 x 105.
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References [CRA 84]
CRANK I. J., Free and Moving Boundary Problems. Clarendon Press, Oxford, U. K., 1984.
[VOL 80]
V. R. VOLLER AND M. CROSS. Accurate solutions of moving boundary problems using the enthalpy method. Int. J. Heat Mass Transfer, 24,P. 545-556, 1980.
[ELG 96]
EL GANAOUI, M. et a/. Stiff problems in Thermofluid dynamic using finite volume method. In Finite volume for complex applications. Eds. F. Benkhaldoun and R. Vilsmeier, Hermes, 1996, p. 717-724.
[MOR 99]
D. MORVAN, M. EL GANAOUI, AND P. BONTOUX. Numerical simulation of 2D crystal growth problems in a vertical BridgmanStockbarger furnace, latent heat effects and crystal-melt in interface morphology. Int.]. Heat Mass Transfer, 42, p. 573-579, 1999.
[ELG 99]
EL GANAOUI M., BONTOUX P. ET MORVAN D. Capture d'un front de solidification en interaction avec un bain fondu instationnaire. C. R. Acad. Sciences Paris, t. 327, Serie II b, p. 41-48, 1999.
[EYM 97]
EYMARD R., GALLOUET TH., AND HEREIN R. Finite Volume Methods, in Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions (Eds), North Holland. Preprint 1997 (to appear).
[LEO 90]
LEONARD B.P. AND MOKHTARI S. Beyond first-order upwinding the ultra-sharp alternative for non oscillatory steady state simulation of convection. Int. J. Numerical Methods Engineering, 30: 729-766, 1990.
Integrating finite volume based structural analysis procedures with CFD software to analyse fluid structure interactions Marcus A Wheel, Andrew Oldroyd, Thomas J Scanlon and Pan Wenke Department of Mechanical Engineering, University of Strathclyde, Glasgow, Gl 1XJ, UK ABSTRACT A finite volume based procedure for predicting deformations in statically and dynamically loaded two dimensional structures is briefly presented in this paper. The paper goes on to describe the integration of this procedure within an existing finite volume based CFD code to provide a facility for analysing fluid structure interaction phenomena. The integrated facility has been used to analyse a flow induced vibration problem in which a moving fluid induces oscillations in a bluff flexible structure around which it is flowing.. Preliminary results obtained from this analysis are included in the paper. Key Words: finite volume method, structural analysis, fluid structure interaction.
1. Introduction Numerical structural analysis is dominated by the finite element (FE) method because of its geometric versatility that allows it to represent the complex geometries of loaded engineering structures. In computational fluid dynamics (CFD) an alternative technique the finite volume (FV) method is popular because it automatically satisfies the conservation principles of fluid mechanics on a local scale. Conventional FV procedures have however lacked the geometric versatility associated with the FE method. Although FE and FV methods have been successfully employed in structural analysis and CFD applications respectively there are many engineering analysis problems in which complex mechanical systems interact with a contained or surrounding fluid. Typical examples include flow induced vibration, wave loading of marine structures and sloshing of liquids in containers. Forming processes such as casting also involve interactions between solid and fluid phases. In these typical interaction problems the deformation of the structure is intimately coupled to the motion of the fluid. In order to analyse these problems in detail a facility with numerical structural analysis and CFD capabilities is required. Existing FE and CFD software packages could be used concurrently to provide such a facility but the two packages would have to exchange information through data files. For a typical transient interaction problem the effort involved in continuously acquiring and storing data in these files will be computationally intensive and the whole analysis
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procedure will therefore be inefficient. One alternative approach to producing a facility for analysing fluid structure interaction problems would be to develop entirely new software with both structural analysis and CFD capabilities. A second alternative would be to develop procedures that could be integrated with existing software in the form of user defined functions to provide the additional features required for the analysis of interaction problems. The recent development of geometrically versatile FV based numerical structural analysis procedures [DEM 94, BAI 95, WHE 96] has enabled both of these alternative approaches to be pursued. The FV based structural analysis procedures exploit features employed in FV based CFD software. Most importantly, the unknown displacement variables are located at the centres of the cell that are used to discretize the structure. Individual cells can be multifaceted and cell geometry is not restricted to a number of predefined shapes as in the conventional FE method. Unstructured cell meshes can be assembled to represent the complex geometries of loaded engineering structures. Although these procedures were originally developed to predict small strain deformations in linear elastic structures there capabilities have subsequently been extended to analyse problems involving nonlinear materials [TAY 95] and large deformations [PAN 99]. 2. A finite volume procedure for predicting deformations in loaded structures To predict deformations in two dimensional loaded structures the FV procedure divides or discretizes the structure into a series of interconnected elements or cells. The geometry of individual cells is unrestricted, each cell can have an arbitrary number of edges and traction forces act on these edges. A typical cell is shown in figure 1. The resultant force found by integrating the traction forces acting on the edges of a particular cell is either equal to zero, in which case the cell is in equilibrium, or equals the rate of change of momentum of the cell. Mathematically this condition can be expressed as
Here, S and V are the surface area and volume of the cell respectively, GJJ, nj and U; correspond to the surface stress, surface normal and cell centre displacement components and p is the material mass density. The finite difference approximation
is used for the temporal discretization of the momentum term on the right hand side of equation 1. The superscripts m-1, m, and m+1 refer to previous, current and future time steps while At is the time step size. To minimize time damping in the numerical procedure the temporal weighting
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is applied to the resultant force term on the left of equation 1. The weighting parameter 9 can be set within the range 0 < 9 < 1. By using a suitable constitutive relationship the stress component terms within equation 3 can be replaced by terms involving the strain components and constitutive properties. It is assumed that the strain components are constant on each edge of the cell so these can be approximated by linear displacement variations involving the unknown cell centre displacements [WHE 96]. On substituting these approximations for the strain components into the resultant force term and equating the resulting expressions to the discrete momentum term a system of algebraic equations containing the unknown cell centre displacements is produced. When 9 is set to zero the unknown displacements at the future time step are given explicitly in terms of the displacements evaluated at the current and previous time steps. The time step size will however be subject to an appropriate restriction. If 9 is non zero then the algebraic equations are implicit and the future displacements must be obtained by solving the equations simultaneously. To validate the FV based structural analysis procedure briefly summarized here a test problem consisting of a unfilled 0.5 m square box supported internally by a 0.0179 m thick cantilever beam was selected. The thickness of the box wall was twice that of the cantilever. The Young's modulus and density of the box material were 5.0 MPa and 500 kgm"3 respectively. Although the majority of the mesh representing the box and cantilever consisted of regularly arranged quadrilateral cells, as shown in figure 2, the intersection between the box and the cantilever was represented by an unstructured mesh of triangular cells, as illustrated in figure 3. A tensile stress of 1 Nm"2 was applied to the lower half of the right outer face of the box and a corresponding compressive stress was applied to the upper part of this face. This induced a state of pure bending in the internal support beam. The computed vertical static deflection of a point located midway up the intersection between the right end of the cantilever and the inner surface of the box was 2.26 mm. The theoretical value for the end deflection of a cantilever of the same length loaded by the same moment is 2.32 mm. The static loads were then removed and the structure allowed to oscillate. The subsequent time dependent deformation of the block and cantilever were computed using the FV procedure with the temporal weighting parameter 9 set to 0.25. The time step size was set to 0.04 s. The variation in the vertical displacement of the point midway up the intersection with time is shown in figure 4. It can be seen from this figure that the numerical procedure does not suffer from time damping because the amplitude of the computed oscillations does not decay with time. The average peak to peak elapsed time is 4.18 s implying that the structure is oscillating at a frequency of 0.2392 Hz. A prediction of 0.2396 Hz was obtained for the first natural frequency of the structure from a modal analysis performed using the commercial FE code ANSYS with the same mesh geometry and material properties. The accuracy of the
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Finite volumes for complex applications
deflection and frequency predictions demonstrate the suitability of the FV procedure for calculating static and dynamic deformations in loaded structures. 3. A CFD analysis of laminar vortex shedding The commercial CFD code PHOENICS was used to simulate laminar vortex shedding around a square bluff body within a confined channel flow. The geometry of the flow domain is shown in figure 4. The aspect ratio of the block, A/B, was fixed at 1.0 and the blockage ratio, B/H, was set at 0.25. All dimensions were normalized with respect to B which was set to 1.0. Dimensionless density and inlet velocity were both set to 1.0 and pressures, velocities and time were all normalized with respect to these specified values. A Reynolds number of 600 was specified to ensure that the flow was predominantly laminar. The computational domain was constructed using a multi block fine grid embedding technique in which the cell mesh was progressively refined in the region around and immediately downstream of the block. This aimed to ensure that the details of the vortex shedding process would be simulated accurately. The fine grid embedding necessitated the use of a collocated grid with pressure and velocity variables stored at the cell centres. An upwind convection scheme was also used. To initiate vortex shedding an initially asymmetric flow field was required. This was generated by performing a steady state analysis with the wall velocities of the block fixed at 1.0 in a clockwise sense. Previous studies demonstrated that the vortex shedding behaviour observed in the subsequent transient analysis was insensitive to the magnitude or sense of the wall velocity prescribed in the steady state analysis. A time step size of 0.04 was selected to perform the transient analysis. The computation was performed using 1875 time steps on a Pentium 200 MMX personal computer with 64 MB of memory. Total run time was approximately 5 days. Pressure was monitored at a point located close to the upper rear corner of the block. Figure 5 shows how the predicted pressure at this location varied with time. The Strouhal number obtained from measuring the average peak to peak elapsed time was 0.1852 for this case. This compares extremely well with an experimentally obtained value of 0.182 for these conditions [DAV 84] and represents an improvement on a previously obtained computational result of 0.16 obtained using an upwind convection scheme on a staggered grid [SCA 99]. 4. Response of the block to fluid pressure loading At each step in the transient CFD analysis the static gauge pressure field around the block was recorded. The complete time history of the pressure field was then applied to the FV based model of the box and cantilever test structure discussed previously. The response of the structure to the pressure loading is shown in figure 5 in which the variation in the vertical displacement midway up the intersection between the cantilever and the box with time is depicted. This response displays a
Fields of application
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characteristic 'beating' behaviour with the amplitude of oscillation increasing and decaying at regular intervals. A Fast Fourier Transform revealed that the two frequency components present in this response correspond to the frequency of the oscillating pressure field and the first natural frequency of the structure. The beating behaviour displayed by the response can therefore be attributed the interaction between these two frequency components. 5. Coupling the CFD and structural analysis The uncoupled analysis discussed in the previous section acknowledges that the motion of the fluid influences the structural response but ignores any effect that the movement of the structure may impose on the fluid. This can only be determined by performing a coupled analysis incorporating the interactions between the fluid and the structure. The FV based structural analysis code was incorporated into the PHOENICS CFD code as a user defined function. Initially, only the flow field is computed in each time step to ensure that periodic vortex shedding is properly established. In each subsequent time step the flow field is computed first. The calculated pressure distribution at the interface between the fluid and the box is then supplied to the structural analysis function and the deformation of the structure determined. The structural deformation is then used to calculate the change in the flow field geometry. The flow field mesh is adjusted to account for any change in geometry. The computation is advanced to the next time step and the adjusted mesh is used to compute the new flow field. The most important feature of this cycle is that all required data is maintained within the physical memory of the PC and continuous access to disk storage is avoided. Currently the cycle does not supply the boundary velocity of the structure to the flow field computation and consequently the transfer of momentum from the structure back to the fluid is neglected. The addition of the structural analysis procedure within each time step added two days to the total solution time. Figure 6 shows the response of the structure predicted by the fully coupled analysis. This response is very similar to that observed when the oscillating pressure field was applied to the structural analysis procedure independently. The beating behaviour exhibited by the earlier response is also observed in the fully coupled response. However, a detailed comparison of the two responses reveals that the maximum amplitude of oscillation is slightly increased in the fully coupled case. When the fluid pressure variation close to the upper rear corner of the box was examined it was found that there was a 2.25% increase in Strouhal number from 0.1852 to 0.1894. This change in Strouhal number represents an increase in the frequency of the forcing function being applied to the structure and, since it is now closer to the natural frequency of the structure, it produces a corresponding increase in amplitude of the structural oscillations. The increase in Strouhal number and amplitude of oscillation also demonstrate that the fluid flow and structural response influence one and other in an intimately coupled way, the details of which can only be modelled by a fully coupled analysis procedure such as the one presented here.
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6. Conclusions A finite volume based procedure capable of predicting the deformation in statically and dynamically loaded structures has been tested on a benchmark structural analysis problem. Comparisons with analytical and numerical solutions have demonstrated that the procedure can accurately predict the static deformation and transient dynamic response of the test structure. The procedure has subsequently been integrated with an existing FV based CFD package. The integrated facility has been used to model the laminar fluid flow around a bluff flexible structure. Vortex shedding from the rear of the structure creates pressure fluctuations within the flow which induce vibrations in the structure. The structural vibrations increase the frequency of the vortex shedding above that observed when the structure is rigid. The increase in vortex shedding frequency is accompanied by an increase in the amplitude of the structural displacement. Future work will focus on improving the analysis by incorporating damping into the structural analysis procedure and including the momentum transfer to the fluid arising from the structural motion. An investigation to examine how the temporal resolution influences the accuracy of the solution will also be conducted. 7. Bibliography [BAI 95]
Bailey, C. and Cross, M, 'A Finite Volume Procedure to Solve Elastic Solid Mechanics Problems in Three Dimensions on an Unstructured Mesh', Int. J. Num. Meth. inEng. 38, 1757-1776, 1995.
[DAY 84]
Davis, R.W., Moore, E.F. and Purtell, L., A Numerical-Experimental Study of Confined Flow Around Rectangular Cylinders. Physics of Fluids, 27, 46-59, 1984.
[DEM 94]
Demirdzic, I. and Muzaferija, S., 'Finite Volume Methods for Stress Analysis in Complex Domains', Int. J. Num. Meth. in Eng., 37, 3751-3766, 1994.
[PAN 99]
Pan Wenke and M.A. Wheel, M.A., A Finite Volume Method for Predicting Finite Strain Deformations in Incompressible Materials, Proc. ECCM 1999, 31 August 3 September 1999, Munich, Germany, 1999.
[SCA 99]
Scanlon, T.J., Stickland, M.T. and Oldroyd, A., 'A Numerical Analysis of Vortex Shedding Within a Confined Channel Flow', to be published in Proc. IMechE Part C, J. of Mech. Eng. Sci., 1999
[TAY 95]
Taylor, G.A., Bailey, C. and Cross, M., 'Solution of the elastic visco plastic constitutive equations: a finite volume approach', Appl. Math. Modelling 19, 747-760, 1995.
[WHE 96]
Wheel, M.A., 'A Geometrically Versatile Finite Volume Formulation for Plane Elastostatic Stress Analysis', J. of Strain Analysis, 31, 111-116, 1996.
Fields of application
Figure 1. A typical multifaceted cell
Figure 2. Cell mesh used to represent box and cantilever structure
Figure 3. Detail of cell mesh at intersection of cantilever and box
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Figure 4. Geometry of flow domain used to model bluff body vortex shedding
Figure 5. Fluid pressure variation downstream of block
Figure 6. Response of structure to fluid pressure loading (uncoupled analysis)
r> fwa
Figure 7. Response of structure to fluid pressure loading (coupled analysis)
A generalized parcel method for the spray dispersion computation
B. Nkonga Mathematiques Appliquees de Bordeaux, Univ. Bordeaux I 351, Cours de la Liberation, 33405 Talence cedex, FRANCE [email protected]
ABSTRACT We propose a generalized formulation of the parcel method for the computation of a dispersed spray in a turbulent flow. This approach take into account an exponential distribution of the radius and the temperature in a parcel of droplets. A system of ordinary differential equations is obtained for the conservative quantities transported by the parcel. The analytical solution of the parcel method system (of ODE) is achieved in a time step with the classical assumption of fixed characteristics of the surrounding flow and for a simplified model for droplets rate of change. In order to improve the methodology on industrial geometries, we have defined reflecting, outgoing and incoming boundary conditions for parcels. The numerical scheme for the gas phase is developed in a finite volume/finite element context for unstructured moving meshes. A generalized Roe solver for multi-species flows and moving meshes is used for the convective flux and a finite element discretization is used for diffusive terms [NKO 97]. The terms coupling the Eulerian and the Lagrangian formulations, as in the classical PIC methods, are obtain by conservative interpolations. Finally, the methodology is performed on the spray injection in a simplified piston engine. Key Words: Multi-phases flow, Mixed Lagrangian/Eulerian Formulation, parcel Method.
Introduction We focus on the behavior of the spray after the complete atomization: the Thin Spray, the Very Thin Spray regimes and the transition to gas phase by
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evaporation. Then we assume that collisions effects and liquid breakup are negligible. Moreover we consider that the spray is composed of spherical drops with uniform temperature. Let us denote /(£,£) the spray probability density function (pdf). In the following analysis, we shall use the phase variable £ = (x, v, r, 9) where x is the position, v the velocity, r the radii and 9 the temperature of the droplet. According to the kinetic model for spray introduced by Williams and the formulations derived in [ORO 81] (24-34) and [CLO 97], we shall use the following Fokker-Planck type equation for the spray: f
+ Vx . (v/) + Vv - (v/) + ~ (rf) + jj ((?/) + Vv - (DegVv/ ) = 0 (1)
where v is the particle acceleration and Dfg is the diffusion coefficient associated with the deterministic model of the surrounding gas turbulence effects. Generalized parcel method
Let us assume that the gas flow is not a function of microscopic variables £ and decompose the function / as a sum of numerical particles fi solutions of the kinetic equation (1). We also assume that there is no interaction between numerical particles and no correlation between the radius distribution, the temperature distribution and the other distributions. The numerical particle is then defined by:
where
and
The behavior of a numerical particle (£} is then defined by its number of droplets NI = (l,/^) , its total mass Me(t), its mean radii ri(t), its mean temperature T t ( t ) , its mean position X^, its mean velocity V^, its total energy Ei(i] = Mi(Ki + CvTi) and its characteristic size Si = \/o^ — X^ • X^:
where me is the mass of a droplet with a radius one, Cv is the specific heat at fixed volume and, for a given function Y, we have:
Fields of application
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Proposition 1 Let us assume that droplets cannot collapsed or break-up, then for a given parcel I, the number NI of droplets is constant and the behavior is defined by the following differential equation (ODE):
where, for any functionY: {Y} t = ?£ (r3Y, f t ) ^
and [Y]t = (Y^t)r / (r 3 ,^),
We denote by Q the differential operator defined by equation (1). This proposition is obtained by computing (Y(x, v,r), Gft)f for the values of F(x, v,r) = 1, Let us denote W the molecular weight, p the density, p the pressure, Y the mass fraction, T the temperature, /z the gas shear viscosity, A the thermal conductivity , Tie the Reynolds number and Pr = 0.75 the Prantl number. We assume that the diffusion process and the energy balance control the exchanges rates. Then the exchange rates of the kinetic equation are denned as functions of the transfer number B, the Nusselt number A/'u, and the averaged temperature on the drop surface Ts. The three parameters (B,J\fu,Ts) are denned by the following non linear system [ORO 81] (94):
where the subscript (d) is used for the drops, (g) for the gas, (v) for the vapor species (produced by drop evaporation), a is the void fraction ( volume occupied by the gas), R is the perfect gas constant, we chose Mud = 1, p* is the normalized equilibrium vapor pressure denned by the Clausius-Clapeyron equation, and Y* is the averaged vapor species mass fraction over the drop's surface,
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where the subscript (i) is used for the inert species parameters. The Reynolds number and the latent heat of vaporization (£(T)) are given by:
u is the gas velocity and u' his fluctuation. Then, the rate of drop radius change (r), the rate of temperature change (#) and the mean aerodynamic drag force (vd r a g ) , are given by:
where
The fluctuations of gas velocity produced a external force for the droplets. This force is defined by:
where u' is the gas velocity fluctuation. Let us denote a a vector of random numbers, a = (01, ..a-,-, ..a^) with — 1 < Q.J < I. We assume that the fluid turbulence is isotropic and Gaussian distributed. Then, for a given vector a the turbulence force in a time step is approximated as follows [DUK 80] (240-241):
where k is the turbulent kinetic energy and r the turbulence time scale. In practice r is of order At. Then the rate of change v, taking into account the jet propulsion effects, the generalized Archimedes, the gravity, the mean drag and the turbulence forces, is defined as follows:
Fields of application
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where [i is the coefficient of the jet propulsion effect, u is the mean flow velocity around the particle, and
In order to develop a numerical method for parcel of drops, we assume that during a time step (t — s) the parameters B, j3, /u, C and g are constant. Then we can compute the analytical solution of the generalized parcel method system. Proposition 2 Let us consider an exponential distribution for the radii:
where
We suppose
coefficients C (drag), K (evaporation), u (jet propulsion), g (external forces) and the surrounding gas velocity, are constant. Moreover, the status of the numerical particle is supposed known at the time s. Then [rp]e = ctprp~ and the behavior of the numerical particle is defined by:
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Finite volumes for complex applications
Oip = j(fj; furtheremore, we can note that a4 = 1, so that TI is the Sauter mean radius. Therefore,
We finally obtain the following simplified system:
Then, the proposition is obtained by the integration of the previous system of equations, when using the relations:
and
Note that lim HKS = e ^, so that at the limit when K —> 0 we recover the solution of the non vaporing spray. We can also obtain, by an analytical integration, the behavior of the size Si of the numerical particle. Previous investigations [DOM 97] have focus on the case of a Dirac radii distribution. In this new context, the Dirac radii distribution can be viewed as a particular case where ap = 1 for all p. Some aspects of this analytical solution was taken into account in the numerical approach proposed by P. O'Rourke [ORO 81]. K
>0
Numerical model
We assume that the motion of the Newtonian fluid phase is governed by the Navier-Stokes equations for time dependent spatial domain. Let us deT note W = \F p, pu, E 1 the conservative variables in the gas phase, Pf —
Fields of application
r
-* EI Mi, Mi Vi,
IT
581
the conservative variables for a numerical parcel and
B(Xi,St) the approximated ball in which live a computational parcel. The equations for vaporizing liquid spray, with the parcel method formulation writes as:
where f is the convective flux, Q and A are the functions describing the parcel motion. Note that if we add a diffusive terms in the gas equation or use a multi-species formulation, there are minor change in what will be developed. j^ i j For a parcel i setting in the mesh element T£ 2 at the time £ n ~ 2 , the characteristics of the surrounding flow are defined as an average W™( (at the time tn) of the values on the vertices (cells) of r" 2 . Assuming that those characteristics and the physical parameters (drag coefficient, evaporation rate, ...) are constants and for a given initial condition, the ordinary differential describing the behavior of the computational parcel is integrated exactly. We rewrite the numerical scheme for spray as follows:
As we assume constant flow characteristics, the stability and the accuracy of the coupled system (gas-spray) is insured only under a CFL type condition [ORO 81]. Parcels localization and boundary conditions Let us consider a parcel with a previous location X£ 2 associated to the mesh element r™ 2. When the new position Xe 2 is computed we define a set of elements T™~2 'n
5
approximating the domain recovering the parcel
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Finite volumes for complex applications
trajectory during the time step:
n— 2-
where r^i = ri inequality:
and for strait lines TIJ and TIJ+I are related by the
Therefore, The construction of ' achieved by a recursive algorithm. For a given element TIJ the space domain is decomposed in four unrecovered zones defined by: the element T, the domains fii, f&2 and f)s. If Xt 2 G TIJ then rg^ — TIJ and the sequence is achieved. If Xe 2 G r^ and there exist an neigbhor element TJ of TIJ such that Ti G f^ then TIJ+I = Ti. Otherwise, the trajectory is crossing the physical boundary. The segment SH associated to rgj and fi; is on the boundary of the physical domain associated to a boundary condition that can be a reflection ( elastic or non elastic) or an open boundary. For reflecting conditions, the trajectory is modified, a new position is defined and then the algorithm is performed with this new position. For an open boundary, the parcel is just deleted from the actual list. This construction set out a natural way to applied the boundary conditions for parcels "crossing" the physical boundary. According to the CFL condition the construction of 7^ 2> 2 is cost-less. Indeed, it is in general y~l
.1. J~l~\-
i
n--
included in the set of the adjacent elements to the nodes of rt 2. A mixed finite volume/finite element formulation is used for the discretization of the gas equation. The representation is "vertex centered" and therefore cell-averages are stored at the vertices of the mesh and the control volumes are the dual cells. Let us denote (Ci(t))i=1 ns a set of non-intersecting cells (dual cells). The integration of the gas equation on the space-time volume gives the following numerical scheme:
where a™ is the volume of the cell Ci, 3^ 'm a conservative approximation of the convective flux with the accuracy of order m = 1 or m = 2, $? the source term produced by the gas-parcel interaction, W™ are components of Wn, P™ components of Pn~5 and APn = p n +2 _/>"-£.
2
are
Fields of application
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Let us denote V(i) the set of the neighboring cells and T(i) the set of the neighboring elements of the cell d. Then $f and $f are defined as follows for the first order finite volume scheme:
The accurate computation of 9^ is costly. When i is a vertex of the triangle r and the parcel is defined by an element r at the time tn+% (X™ —77
I i,
2
— ft 1
we can use the approximation (OTe ~ 9i£ ), where Qit
2
2
G T] then
i
is the barycentric
->n+-2
coordinate of Xt associated to the vertex i in the triangle. Otherwise, we can assume that Q?t ~ 0. The fluid step is resumed by the following system:
This system is linear (diagonal) for explicit schemes (p = n) and non linear for implicit scheme (p = n + 1). In practice we use, in this last case, a linearized implicit scheme obtain as follows:
dW
wr
The linearized implicit scheme is then reduced to the resolution of a non symmetric linear system. The resolution of this scheme is in general achieved by Gauss-Seidel relaxation method. It is the costly part of the implicit algorithm.
Application: Injection in a piston engine Let us focus now to the piston engine application. In this case, the jet(s) position(s) and characteristic are assumed known. The fluid initially at rest is compressed by the piston motion in the Ricardo prechamber. Then the spray is injected at the position defined by Xj nj -(t), with a mean velocity Vinj(t) — 115 m/s and a mean drop radius rinj = 4 * 10'6 m. The injection starts after 27ms and ends at 28ms with a total mass of Q.8g injected. A total number of 6060 numerical parcels are injected. The numerical behaviour of the spray and the boundary conditions are validated in this context (Fig 1 ). Computations has been performed with a PVM parallel code and the figures are ploted in the case of 4 sub-domains partition.
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[CLO 97]
CLOUET, J.F. AND DOMELEVO, K. Solution of a kinetic stochastic equation modeling a spray in a turbulent gas flow. Math. Models Methods Appl. Set., 7(2):239-263, 1997.
[DOM 97]
DOMELEVO, K. AND SAINSAULIEU, L. A numerical method for the computation of the dispersion of a cloud of particles by a turbulent gas flow field. J. Comput. Phys, 133(2):256-278, 1997.
[DUK 80]
[NKO 97]
JOHN K. DUKOWICZ. A particle-fluid numerical model for liquid spray. Journal of Computational Phisics, 35:229-253, 1980.
B. NKONGA. On the conservative and accurate cfd approximations for moving meshes and moving boundaries. To be published in Computer Methods in Applied Mechanics and Engineering.
[ORO 81]
PETER J. O'RouRKE. Collective drop effects on vaporizing liquid spray. PhD Thesis, Los Alamos National Laboratory, 83, November 1981.
Figure 1: Spray density evolution and Velocity field at 23.06ms : 6060 total Particles, Implicit scheme, moving mesh of 4202 vertices.
Finite Volume Methods for Multiphysics Problems
C.Bailey, M.Cross, K.Pencleous, G.A.Taylor, N.Croft, D. Wheeler, H. Lu Centre for Numerical Modelling and Process Analysis University of Greenwich Woolwich, London SE18 6PF, UK
ABSTRACT There is a growing need for accurate and robust multi-physics based simulations tools that have the ability to model fluid flow; heat transfer, possibly with solidification or melting; and solid deformation in an integrated manner. Engineering Processes are now benefiting from the use of such tools where potential defects are being identified at an early stage in the design process. This paper details a finite volume formulation and solution strategy for predicting the physics associated with solidification based processes. Key Words: Multiphysics Modelling; Solidification; Casting; Electronic Packaging
1. Introduction
A number of real industrial processes are governed by interacting physical laws, where the materials involved in the process may be both solid and/or fluid. An example of this is the casting of metal components where the metal, originallyabove its melting point, is placed into a mould cavity then allowed to cool and solidify into its required shape. The challenge for the modelling community, provided by this type of problem, is to capture all the relevant physics taking place and ensure that there interactions are properly simulated using a single modelling framework. Although finite volume methods are very well established for fluid flow phenomena, the dominant discretisation technique for solid mechanics (hence stress) is the finite element method. Lately, finite volume methods have been used to analyse solid mechanics problems on unstructured meshes [DEM 95],
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Finite volumes for complex applications
[BAI 95], [TAY 96].These methods have proved to be as accurate as classical finite elements for both linear and non-linear materials. Integration of this type of stress analysis within a finite volume CFD framework provides the capability of solving fluid-structure interation problems using a single computational framework. 2. Governing Equations
2.1 Fluid Flow The general equations for the conservation of momentum and mass are:
where u is the velocity vector and p,, p, and p are the dynamic viscosity, density and pressure at time t. The source term 5 contains the buoyancy and Darcy terms due to temperature changes and solidification. 2.2 Temperature The conservation of energy is given by
where h, k and T are the enthalpy, thermal conductivity and temperature. L is the latent heat, which is the energy released due to solidification, and fi represents the liquid fraction (1 — liquid; 0 = solid). 2.3 Solid Mechanics The incremental equilibrium equations are :
Using Hookes Law the incremental stress,
is given by:
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where [D] is the elasticity matrix and Ae, Aet and Acvp are the total, thermal and viscoplastic incremental strains respectively. [L] is the differential operator matrix and Ad are the displacement increments.
3. Discretisation
Figure 1 details two types of control volume that can be used in the finite volume discretisation procedure. Cell-centred control volumes are the same as the mesh elements used to represent the domain. Vertex based control volumes
Figure 1: Control Volumes are built up around each node where the boundary for each control volume is the face that joins the element centre to the centre of the element faces. 3.1 Cell centred For the cell centred approach, velocity components, temperature, pressure, and liquid fractions are stored at the centres of each mesh element making up the control volume. The general form of the conservation equations for mass, momentum, and energy can be written as:
where, from left to right, the terms represent the transient, convective, diffusive and source contributions respectively. The dependent variable is represented by
588
Finite volumes for complex applications
0 which is unity for mass, the velocity component for momentum, and enthalpy for energy. Integrating the above equation over each control volume and using the divergence theorem gives:
Linear approximations are used for the transient term and source terms over the control volume. The diffusive and convective terms are then approximated at the faces of the control volumes. This gives:
where the summation is over the number of faces. „, Af and dap are the dependent variable at neighboring control volume, face area, and distance between centres of neighboring cells. To obtain accurate values of the diffusion coefficient, Tf, at the face the harmonic mean is used. For pf, in the convection term, upwinding is used and to evaluate the normal component of the velocity at the face, (u.n), the Rhie-Chow interpolation method is used. The value of
3.2 Vertex based Unlike the flow and heat equations, above, the stress equations are inte grated over vertex based control volumes. Integrating the equilibrium equa tions over a vertex-centred control volume and using the divergence theorerr gives:
where the incremental stress components, Acrjj, are approximated at the faces of each vertex based control volume. Using shape functions, the displacement at any point within a mesh element can be written as a combination of the values stored at the nodes of that element, therefore:
where [N] is the matrix of shape functions and Av is the vector of vertex stored displacements. The equilibrium equations, integrated over each control volume,
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can now be written in terms of nodal incremental displacements [TAY 96]:
4. Solution Procedure
The above discretised equations for dependent variables at both cell-centred and vertex-centred control volumes can be assembled to provide systems of equations for the dependent variable <j) representing velocities, temperature, fluid pressure, and nodal displacements. The form of each system of equations is: where [A] is the system matrix and b contains source terms. These integrated systems of equations are solved using the following solution procedure: (1) (2) (3) (4) (5) (6) (7) (8)
Solve flow equations using the SIMPLE procedure, [PAT 80] Solve Energy equation for temperatures and liquid fractions [VOL 91] Evaluate other variable quantities, eg. physical properties, etc Repeat steps 1-3 until convergence Solve incremental solid displacements Update total stress variables. crnew = a0id + ACT Recalculate geometrical quantities Repeat steps 1-7 for time-step advancement
Note, in step 7, the geometrical quantities are recalculated due to the changes in displacement from the computational solid mechanics analysis, and the equations in steps 1 and 2 are rediscretised over the updated mesh. 5. Results
Figure 2 shows the computational domain of a silicon chip that is bonded to a printed circuit board using very small solder bumps. This is achieved by passing this assembly through a furnace, heated to 230C, where the solder, in the form of a paste, melts then, as the assembly leaves the furnace, solidifies to form the required joint. Under the solder bumps, copper connections are used to carry data into the printed circuit board. The PHYSIC A modelling
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Finite volumes for complex applications
framework [PHY 98] is used to simulate this fusion process, where the physics of heat transfer (solidification) are coupled with stress to predict the resulting stress profiles in the assembly. Figure 3 shows both the solidification
Figure 2: Computational domain for electronic package. profiles for the solder bumps and the stress profiles throughout the assembly (dark contours representing high stress). Clearly we can see that the thermal miss-match between the materials in the assembly is giving rise to stress concentrations around the copper via connections. The next example illustrates coupled flow, solidification, and stress modelling for the casting of lead ingots. Here molten lead is poured into an iron mould and then allowed to cool and solidify. Figure 3 shows the computational domain for this analysis. A quarter of the domain is shown with symmetry planes at the centre. It is assumed that the mould is instantly filled and that the cast and mould temperatures start at 730C and 300C respectively. As the molten lead cools, thermal gradients will evolve resulting in thermal convection as lead flows from hotter to cooler regions. Figure 4 shows the magnitude of thermal convection after 30 minutes of cooling. Figure 4 also shows both PHYSICA and thermocouple temperature profiles at the centre of the ingot. The affect of latent heat release during the phase change from liquid to solid can clearly be seen. The PHYSICA predictions compare very favourably with the gathered data. Figure 6 (left) shows the solidification profile (dark area being liquid) after 100 minutes of cooling. As expected the lead solidifies inwards from the
Fields of application
Figure 3: Solidification profiles in solder bumps and stress in package.
Figure 4: Computational domain for mould and lead ingot.
591
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Finite volumes for complex applications
mould walls towards the centre of the casting.
Figure 5: Thermal convection and temperature comparison.
Figure 6: Solidification front and stress in mould. As the lead cools it heats the surrounding mould in a non-uniform manner. The thermal gradients occurring in the mould material results in thermal stress. Figure 6 (right) shows the magnitude of stress throughout the mould (dark area being high stress) after 100 minutes of cooling. These type of predictions provide useful information for mould desiginers to minimise stress concentrations and hence cracks in the moulds.
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6. Conclusions The rationale behind the PHYSICA framework is to supply modellers with a tool which enables the development of computational mechanics based models where interactions between the participating physics can be accomplished in a robust and efficient manner. This paper has summarised the discretisation and solution procedures used within the PHYSICA framework. The capability of this modelling approach for real world processes is illustrated for electronic packaging and lead casting. Bibliography [BAI 95]
BAILEY C. CROSS M., A Finite Volume Procedure to Solve Elastic Solid Mechanics Problems in Three Dimensions on an Unstructured Mesh, Int. J. Num. Meth. In Eng., 38, 17571776.
[DEM 95]
DEMIRDZIC I. MUZAFERIJA S., Finite Volume Methods for Stress Analysis in Complex Domain s, Int. J. Num. Meth. In Eng, 37, 3751-3766
[PAT 80]
PATANKAR S., Numerical Heat Transfer and Fluid Flow, Hemisphere, 1980.
[PHY 98]
PHYSICA, A Software Environment for the Modelling of Multiphysics Phenomena: http://phys ica.gre.ac.uk/
[TAY 96]
TAYLOR G.A. BAILEY C. CROSS M., Solution of Elastic/Viscoplastic Constitutive Equations : A Finite Volume Approach, Appl. Math Modelling, 19, 746-760 (1996).
[VOL 91]
VOLLER V. SWAMINATHAN C., General source-based Method for Solidification Phase Change, Num Heat Transf - Part B, 19, 175-189.
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A Finite Volume discretization and multigrid solver for steady viscoelastic fluid flows
H. Al Moatassime S. Raghay and A. Hakim Faculte des Sciences et Techniques, Departement de Mathematiques et Informatique, B.P. 618, Marrakech, Maroc
ABSTRACT A finite volume technique has been introduced in an attempt to simulate a viscoelastic flow. The steady flow of a Oldroyd-B fluid through a 4:1 abrupt contraction has been chosen as a prototype example due to the existence of previous simulations in the literature. The finite volume method (FVM) is used to discretise the conservation and constitutive equations and a multigrid technique is used. The FVM is proven to be quite capable to handle numerically, viscoelastic models with low computational cost. Its use is recommended as a viable alternative to the solution of viscoelastic problems using a variety of constitutive models. Key Words: Numerical simulation; finite volume; planar contraction; viscoelastic flows; Oldroyd-B model; non-uniform staggered grid; multigrid technique. 1. Introduction For the past several years, numerical simulation of viscoelastic flows has been a powerful tool for understanding the fluid behaviour in a variety of processes of both industrial and scientific interest. Polymeric fluids, owing to their viscoelastic character, are of particular interest in the numerical simulation because of their wide applications in material processing and their behaviour different from that of Newtonian fluids in ways which are often complex and striking. Although there have been many successful numerical complexities that arise due to the change of type, i.e., hyperbolic-elliptic or hyperbolicparabolic. The earlier numerical schemes such as central finite differences [Per 77], Galerkin finite elements [Cro 84] and spectral finites elements [Ber 87] to solve the equations are unstable when the elasticity of the flow becomes
596
Finite volumes for complex applications
significant. While each of them has its advantage and disadvantages, the search for even better and/or faster methods still continues. In that respect, it was inevitable that the finite volume method should also be tried within the viscoelastic context. However, the viscoelastic simulations with FVM are very limited [Sas 95]. Contraction flows of viscoelastic fluids are of importance in fundamental flow property measurements as well as in many industrial applications involving molding and extrusion of polymer melts and solutions. They are also a challenging class of problems for analytical and numerical work. Numerous theoretical studies have been devoted to the study of viscoelastic fluids in contraction flow, and there is a large literature on this subject; see for example ([Bog 87], [Mit 90]). However, the theoretical prediction of entry-flow for nonNewtonian fluids remains a different task. The constitutive equation used to describe the rheology of the polymer has fading memory effects, and may contain nonlinear terms that add to the complexity of the problem. The presence of a geometrical singularity poses another major challenge to the numerical simulation of contraction flows. In this paper, we conduct numerical simulation of the planar contraction flow of viscoelastic fluid. The FAS (Full Approximation Storage) multigrid algorithm is used to simulate large Weisenberg number. 2. Governing equations
For the two-dimensional flow of an incompressible viscoelastic fluid, the governing equations are continuity, the x and y component momentum equations, and three component stress equations from the Oldroyd-B constitutive equation. The solution of these equations, with appropriate boundary conditions, gives u, v, r xx , Tyy, Txy and the pressure p as functions of x, y. The equations are expressed below in terms of dimensionless variables.
Boundary conditions
The boundary conditions employed for the velocity are given below:
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Figure 1: Computational domain for contraction flow In inlet and outlet section : Fully developed Poiseuille flow is imposed
where Uv — \ and Um = ^f The extra-stress components are fixed along the inlet section. They correspond to a fully developed flow between parallel plates.
at the centerline we impose symmetry conditions:
and adhesion at the wall:
3. Numerical method To solve the above coupled non-linear equations at moderate cost, the use of an iterative numerical method has to be made. It has to be noticed that all of the governing equations can be written in the form of the general transport equation as follows:
where <£ is the working variable which can be a component of a vector or tensor and even a constant. The coefficients A and F have different meanings
598
Finite volumes for complex applications
Figure 2: The control volume AV for grid point P for different equations, and 5$ is called the source term which includes all the terms that cannot be taken into account in the convective and diffusion terms, and has different contents for different equations.
A simple finite volume formulation is used for the spatial discretisation. The flow domain is divided into a set of control volumes AF around P with bounding surface area A as shown in Figure 2. Integrating equation (1) over the control volume AV and using the divergence theorem, we have
where 5$ is the integral of the source term 5$. This can be linearized as
Fields of application
599
in which Sc is the part of the S$> that does not explicitly depend on 3> and Sp is the coefficient of 4>p,which is made negative to enhance the numerical stability of the discretized equation system (2). where, an overbar means that the applied values are evaluated using the known fields for iteration level n. By using a proper spatial variation approximation scheme, the final discretized equations relating the 3>p to its neighbouring grid point values can be expressed symbolically in a general form in every control volume:
where
where Dnb and Pe are respectively the diffusion coefficient and the Peclet number. sign(nb) — + when nb £ {w,s} and sign(nb] = —when nb e {n,e}. The summation is to be taken over all the neighbouring grid points nb of the central point P, and the coefficients anb are the functions of the working variable, and their structures depend both on the approximation scheme used and the form of the cell. These coefficients determine the spatial accuracy of the final solution. In our calculations, the Hybrid scheme proposed by Patankar [Pat 80] is employed to solve momentum equations,
while Upwind scheme is used in the constitutive equations for the stresses.
In viscoelastic flows the convective terms are too small in comparison with the source terms and special treatment of the source term is needed. The computation of these terms requires the first gradient of Txx,Tyy and rxy. The term drxx/dx, as like as the other terms, is assuming quadratic variation of rxx along the x-direction. Thus , drxx/dx is written as 2ax + b. Solution method
To obtain the kinematic fields, an equation for the pressure is obviously necessary because it is also an unknown. The strategy of pressure correction is utilized to produce the pressure equation, in which the continuity of the field is enforced via a pressure correction so that the resulting pressure relation, which couples the pressure and the velocities, replaces the continuity relation while the momentum equations retain their role for determining the velocity field. To avoid physically unrealistic field, such as the checker-board velocity and pressure distribution, staggered control volumes have to be used. As shown
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Finite volumes for complex applications
in Figure.3, for the P-centered control volume for scalar fields, the velocities Uk (k = x,y)are discretized using their values on the faces normal to the k direction; thus, the location of the control volume for Uk in the momentum equations is staggered only in A; direction relative to the control volume for the fields (cf. Figure 3). The steps of present algorithm are as follows : * * * *
Computations of velocities. Computation of pressure and correction of velocities Computation of stresses. Convergence control and return to first step if necessary.
Multigrid Procedure After discretization we get a non linear finite dimensional operator Ln and we have to solve Ln(u,p,r) = bn In order to solve this problem we use a FAS (Full Approximation Storage) multigrid algorithm. Let (^fc)i
by a relaxation procedure Mk and we denote by 72.™( w fc^>Pfc^> T A:^) the vector obtained after ra iterations with the initial vector (uk^l\pk^ ,Tk^}The object of the FAS algorithm is to solve
on the finest grid fJn by solving intermediate problems (4) on coarser grids in order to reduce the number of iterations on the finest grid.
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Figure 3: Two-dimensional staggered mesh and the control volumes for u,v,p well as r FAS algorithm {0} (un,pn, rn} initial approximation of solution of (5) {1} k\=n {2} (uk,pk,Tk) = M™k(uk,pk,Tk) (mk iterations of relaxation on grid ftfc) {3} Compute 6 fc _i = 1^1^(bk - Lk((uk,pk,rk)) + Lfc-i^- 1 ^,^,-^)) (w*-i,pfc-i,Tjfe_i) = nl~1(uk,pk,rk) (initializing of solution on grid Slk-i) k := k — 1 (crossing to the coarser grid) {4} I f f c > 1 goto {2} Else (mi iterations of relaxation on the coarsest grid fii) {5} Correct the approximation obtained on flk+i with the extension of
k := k + 1 (crossing to the finer grid) {6} (uk,pk,i~k) — M ^ k ( u k , p k , T k ) ({J>k iterations of relaxation on grid f^) {7} If k < n go to {5} Else convergence test If convergence done, good, end Else go to {2} In practice for the FAS procedure we take mi = 10, /u=3 and mk = 2(k = 2, • • •, n) for small Weissenberg numbers and mi = 50, p,-20 and mk = 2Q(k = 2, • • -, n) for large Weissenberg numbers.
4. Numerical Results
The flow along the centreline has an elongational nature, and the elongational properties of the rheological models play an important role in determining the way it responds to the accelerating forces when it approaches the corner. Velocity profiles for Weissenberg number We — 1 are plotted in Figure
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Finite volumes for complex applications
4. This figure shows an important velocity overshoot near the coner. Marchal and Crochet [Mar 87] have also predicted similar velocity overshoot and high stress peak near the singular region in their high De calculations with Oldroyd-B fluid. First normal stress difference profiles for different values of the Weissenberg number are reproduced in Figure 5. The maximum of (TXX — ryy} is reached just upstream of the entry section. As We increases, the value of this maximum increases sharply. Profiles of the normal extra-stress rxx as well as rxy are presented in Figure 6. This figure shows a high rxx and Txy peak near the corner. In Figure 7 we present distribution of rxx and rxy for (Re, wr, We) — (0.01,0.8,1). It is clear that the re-entrant corner represents a singularity, many lines pass a cross this point. Conclusion The planar 4 : 1 contraction flow of Oldroyd-B fluid has been simulated by utilizing a finite volume method with multigrid technique on a non-uniform staggered grid system. With FVM the velocity and pressure fields are made to satisfy one the same momentum equation at the end of each step, and the pressure correctors are used to correct the velocity field only. To make the method suitable for viscoelastic flow computations, decoupled techniques are employed, and artificial diffusion terms are introduced on both sides of the discretized constitutive equations to stabilize the numerical calculation. With the method, thanks to the modest demand on memory, it becames possible to solve large problems on small computer systems. As a results, the present numerical simulation have allowed us to reproduce such of the experimental results. Bibliography [Per 77] M.G.N. Perera, K. Walters, Long-range memory effects in flows involving abrupt changes in geometry. Part I. Flows associated with L-shaped and T-shaped geometries, J. Non-Newton. Fluid Mech. 2 (1977) 49-81. [Cro 84] M.J. Crochet, A.R. Davies, K. Walters, Numerical Simulation of Non-Newtonian Flow,Elsevier, Amsterdam, 1984. [Ber 87] A.N. Beris, R.C. Armstrong, R.A. Brown, Spectral/finite element calculations of the flow of a Maxwell fluid between eccentric rotating cylinders, J. Non-Newton. Fluid Mech. 22 (1987) 129-167. [Bog 87] D.V. Boger,Ann. Rev. Fluid Mech., 19 pp. 157-182, 1987.
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Figure 4: U and V on y = 0 and y = 1 respectively
Figure 5: (rxx — ryy) on y = I for We — 0.8,1 and 1.2 [Mit 90] E. Mitsoulis, Numerical simulation of viscoelastic fluids, in : N.P. Cheremisinoff (Ed.),Encyclopedia of Fluid Mechanics, vol. 9, Polymer Flow Engineering, Gulf Publishing Company, Dallas, Texas, USA, 1990, pp. 649-704. [Pat 80] S.V. Patankar, Numerical Heat Transfer and Fluid Flow,McGrawHill, New York, 1980. [Sas 95]
G.P. Sasmal, A finite volume approach for calculation of viscoelastic flow through an abrupt axisymmetric contraction, J. Non-Newton. Fluid Mech. 47 (1995) 15-47.
[Mar 87] J.M. Marchal and M.J. Crochet,J. Non-Newton. Fluid Mech. 26 (1987), 77-114.
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Figure 6: rxx and rxy on y — 1 respectively
Figure 7: isolines of TXX and rxy
respectively
Complexity, Performance and Informatics
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Various CG-type methods applied to finite volume scheme
0. Schmid 1 DLR, Cologne, Germany A. Bufimann, E. von Lavante and M. Moczala University of Essen, Essen, Germany
ABSTRACT This paper compares comprehensively preconditioned classical, Lanczos and Amoldi methods for a subsonic turbine cascade. The well known symmetric Gauss-Seidel (SGS) is selected, representing a classical method. General minimal residual method (GMRES) is chosen as a member among the Arnoldi methods and Bi conjugate gradient stabilized (BCGSTAB) represents a truncated non-symmetric Lanzcos method. Two preconditioners are tested. The first one is the prolific diagonal incomplete lower upper (DILU) preconditioner with zero fill-in. The second one is the lower upper symmetric Gauss-Seidel (SGS) preconditioner. Key Words: Computational Fluid Dynamics, Navier-Stokes Equations, Finite Volume Method, Implicit Method, Upwind Shemes, MUSCL-Extrapolation, CG-type Methods, Classical Methods, Steady Flow, Convergence, Viscous Flow, Iterative Solution.
1. Introduction The numerical simulation of typical turbomachinery components, such as compressor and turbine rows or stages, requires excessive amounts of computational resources. This is not only due to the highly complex physics of the flow, which makes very fine computational grids necessary, but also because of the need for high timewise accuracy.
presently University of Essen, Essen, Germany
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Finite volumes for complex applications
The flow in these components is almost always unsteady; however, at times, it might be admissible to filter out the time dependent part of the flow. Under this assumption, the flow can be considered steady, and the computation might converge to a steady state result. In this case, one is interested in accelerating the rate of convergence in a way that is not necessarily time accurate, marching in a properly defined pseudo-time. The problem of acceleration of a numerical scheme for solving the compressible Navier-Stokes equations in complex components of turbomachines can be approached by implementing an appropriate implicit procedure. The implicit operator should provide fast convergence to steady state results for the steady computations. Numerous different implicit algorithms for solving the Navier-Stokes equations have been published to date. These include, for example, the work of Earth [BAR87], Jameson [JAM91], Pan and Lomax [PAN86] and Wright et al [WRI96], to name just a few. What was missing, in our opinion, is a comprehensive comparison of the different implementations of implicit solvers with regard to their ability to predict complex flows in modern turbomachinery, such as highly loaded turbines or compressors. In the present work, several implicit methods were obtained by using different approximate inversion procedures of the same descrete delta form of the governing Navier-Stokes equations. These were applied to numerical simulations of several simple test cases as well as the experimentally tested and documented T106 turbine cascade. The computed data were compared with available theoretical or experimental results.
2. Basic Algorithm The governing equations are the two dimensional Navier-Stokes equations in divergence form written as
where Q is a vector of dependent conservative flow variables and F and G represent the flux vectors. The well known details of this model equation can be found in [JAM91] and [PU85].
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Discretization of equation (1) using first order accuracy in time yields:
where the superscripts denote the time level and the subscripts denote the spatial location. The fluxes F, G, Fv and Gv are the time properly denned numerical fluxes, obtained from a particular spatial discretization. 6$ and 6^ are the difference operators acting on the numerical fluxes. In the present work, the spatial discretization of the governing equations was carried out by Roe's Flux-Difference Splitting procedure [ROE84]. The interpolation slope of the primitive variables was limited by the modified Albada limiter, according to the previously published MUSCL type procedure (see, for example, [EVL94]). The implicit operator is defined in the delta form of equation (2), obtained by a proper linearization of the numerical fluxes F, G, Fv and Gv. It can be written, after splitting the Jacobians of the flux vectors A = |£ and B — |§ into a matrix with non-negative eigenvalues A+ and non-positive eigenvalues A~, as
In the above equation, M and JV are the Jacobian matrices obtained from the linearization of the viscous fluxes Fv and Gv, respectively. They were centrally differenced and |r, ^7- denote backward and forward differencing respectively. The convective fluxes in the explicit part were linearized by assuming that Roe's numerical damping term was locally constant. Taking, for example, the numerical flux
this results in
with the rest of the Jacobian matrices evaluated correspondingly. 2.1 Inversion of the Implicit Operator Equation (3) can be expressed in a simple form as
where M denotes the implicit operator.
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The solution of equation (5) involves the inversion of the implicit operator matrix M, which is rather large, sparse and linear. A non-iterative direct method to invert it would be, among several possibilities, the Gaufi-elimination. However, this method of inversion of M would result in a dense matrix and would therefore require large amounts of computer memory. Worse yet, many fill-ins would have to be computed, making the algorithm slow. Another possible method of inverting M would be one of the approximate factorization (AF) schemes, such as the ADI scheme using the Thomas algorithm. This method is simple and cheap to implement, but introduces a factorization error that effectively limits the CFL numbers to rather small values. It has yet another significant disadvantage: in the case of zonal grid decomposition it requires special implicit treatment of the interzonal communication. With a simple interzonal boundary condition, the ADI scheme induces large errors on the zonal boundaries delaying the convergence significantly. Therefore, the present authors decide to use and test several iterative methods that, in our opinion, offer an acceptable compromise between computational efficiency and complexity. The iterative inversion of M allows updating the interzonal boundary values after each subiteration, providing a simple implicit-like boundary condition. 2.1.1. Relaxation Methods Using equation (5) as a starting point, relaxation methods split the implicit operator in two parts. After some trivial manipulations, one gets the following equation, which describes every relaxation method:
where k denotes the subiteration level. For the Symmetric Gaufi-Seidel scheme (SGS), the matrix P is
Here D is the diagonal part of M, L is the lower part of M and U is the upper part of M. The solution will be reached by a forward and a backward sweep. 2.1.2 Conjugate Gradient Methods CG-type methods use the non-symmetric or the symmetric Lanzcos process to build up the Krylov subspace. After that the residuum is minimized over the given subspace by a Galerkin or a least-square condition. Subiterations are introduced in the following way
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The minimum vector y^ is calculated through a least-square procedure,
or by implementing a Galerkin condition
Without preconditioning, the tested CG-type methods behave very poorly. For this reason, two preconditioners, the distributed incomplete lower upper preconditioner (DILU), and the symmetric Gaufi-Seidel preconditioner (SGS) were implemented. Both preconditioners are well suited for block matrices. All the tested solvers are using the same termination condition for the maximum number of subiterations. The subiteration process is stopped when either the term ~fe r is smaller than 10~2, or the maximum of 500 iterations is reached.
3. Results The above schemes were tested on several two-dimensional configurations possessing increasing degree of complexity. The initial simple test cases were a bump in a channel, a flat plate and a subsonic diffusor. There are well known test cases treated in detail previously and will not be, therefore, discussed here.
3.1 T106 Turbine Cascade The axial T106 turbine cascade has been designed and extensively experimentally tested by Fottner et al. [FOT78] and is therefore well suited for further validation of the present results. The flow in the T106 turbine cascade was computed for an inflow Mach-number of 0.29 using the implicit methods mentioned above. The computational grid, consisting of 5 blocks of grids (zones) is shown in Fig. 1. Here, the body-fitted O-grid is split into two parts, connected by three H-grids. The convergence informations for the implicit schemes discussed in the previous chapter are summarized in the tables below. The resulting parameters shown are: B2, the exit flow angle avaraged over the cascade width; C, the total pressure loss parameter, defined as C = (pt\ ~Pt-2)l(pt\ — Pi}', maximum available CFL-number; the maximum computer RAM requirement and the total CPU time on a SGI-Power Challenge given in hours. Where applicable, the computed values are compared with experimentally measured data. All the simulations were carried out using the k-cj turbulence model.
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In this case, the fastest convergence to 6 degrees of magnitude was obtained using the SGS relaxation scheme. The most effective CG-type procedure was the BiCGSTAB method, combined with the SGS preconditioner, directly followed by the method of steepest descent using the SGS preconditioner. CG methods with DILU preconditioning were not really competitive.
Figure 1. Computational grid for the T106 turbine PA"}
c [%]
26.80 2.08
CFL C P U - t i m e [h] memory [KB]
27.40 8.70 40.0 4.96 1857
27.43 8.72 40.0 4.49 1857
27.38 8.15 40 0 4.60 1857
27.41 8.74 40.0 1.95 1857
Table 1. Relaxation-type schemes SD-t-SGS
/M°i C (%}
26.80 2.08
CFL C P U - t i m e [h] m e m o r y [KB]
| BiCGSTAB + SGS
27.42 8.73 40.0 1.81 1861
| FOM(5) + SGS
27.70 8.72 40.0 1.77 1920
| GMRES(5) + SGS
27.39 8.73 40.0 2.46 2008
|
27.39 8.73 40.0 2.51 2023
Table 2. CG-type schemes with SGS preconditioning SD + DILU
0>A°\ C [%] CFL C P U - t i m e [h] m e m o r y [KB]
26.80 2.08
27.30 8.97 35.0 3.20 1920
|
BiCGSTAB + DlLU 27.40 8.73 40.0 2.47 1979
[ FOM(5) + D I L U 27.39 8.73 40.0 3.15 2067
] GMRES(5) + DILU 27.39 8.73 40.0 3.14 2082
Table 3. CG-type schemes with DILU preconditioning The density contours, shown in Fig. 2, are a clear indication of the relatively simple flow in this configuration. Evident is the viscous wake behind the turbine blade and the boundary layer, thickening on the suction surface towards the trailing edge due to the adverse pressure gradient.
]
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Figure 2. T106 density contours 4. Conclusions
Different iterative schemes for implicit solutions of the Navier-Stokes equations were tested on several configurations from the area of turbomachinery. Several iteration strategies were implemented. The subiterations resulted in implicit-like interzonal boundary conditions. The fully implicit procedures were effective in accelerating the time wise computation, allowing much larger time steps than the explicit formulation resulting in up to six-times faster convergence. For the present configurations, the most effective implicit procedure was the SGS, followed closely by BiCGSTAB with SGS as preconditioner Overrelaxation methods using a relaxation parameter between one and two burst after a few time steps, due to the high relaxation parameter. Every preconditioned CG-type method converges faster than the Gaufi-Seidel method. Preconditioned CG-type methods depending on the truncated nonsymmetric Lanzcos process are faster than those restarting the symmetric Lanzcos process to construct the Krylov-subspace. Nevertheless all of them can not reach the convergence history of the symmetric Gaufi-Seidel method, bringing up to mind that the use of the SGS scheme is really an advisable choise to calculate fluid dynamic problems.
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5. Acknowlegements and Bibliography A major part of the present work was done during the lead authors stay at the DLR, Institute for Propulsion Technology. Many calculations were done with the TRACE-S code developed by the lead author's project leader D.T. Vogel(DLR) who helped the lead author in several tasks. [BAR87]
BARTH, T. J., Analysis of Implicit Local Linearization Techniques for Upwind and TVD Algorithms, AIAA paper 870595.
[JAM91]
JAMESON, A., Time Dependent Calculation Using Multigrid with Applications to Unsteady Flows Past Airfoils and Wings, AIAA paper 91-1596.
[PAN86]
PAN, D. AND LOMAX, H., A New Approximate LU Factorization Scheme for the Reynolds-Averaged Navier-Stokes Equations, AIAA paper 86-0337.
[WRI96]
WRIGHT, M. J., CANDLER, G. V., AND PRAMPOLINI, M., Data-Parallel Lower-Upper Relaxation Method for the Navier-Stokes Equations, AIAA Journal, Vol.34, No. 7, July 1996.
[ROE84]
ROE, P. L. AND PIKE, J., Efficient Construction and Utilisation of Approximate Riemann Solutions, Computing Methods in Applied Sciences and Engineering, VI, pp. 499-516, INRIA, 1984.
[EVL94]
VON LAVANTE, E., HILGENSTOCK, M. AND GROENNER, J., Simple Numerical Method for Simulating Supersonic Combustion, AIAA paper 94-3179.
[PUL93]
Pulliam, T. H., Time Accuracy and the Use of Implicit Methods, AIAA paper 93-3360-CP.
[PU85]
PULLIAM, T. H. AND STEGER, J. L., Recent Improvements in Efficiency, Accuracy and Convergence for Implicit Approximate Factorization Algorithms, AIAA paper 85-0360.
[FOT78]
FOTTNER, L., HAPPEL. H.W. ET AL, Anwendung neuer Entwurfskonzepte auf Profile fur axiale Turbomaschinen, Teil II, Abschlufibericht Zukunftstechnik Luftfahrt 1978 MTU 4.14-2.
[STE96]
STEINERT, W., SCHREIBER, H. A., WEBER, A., Experimente am transsonischen Verdichtergitter, DLR-TSG-89-5, DLR Koln 1996.
A Newton-Relaxation Finite Volume Scheme for Simulation of Dynamic Motion Bruce A. Jolly, Magdi Rizk Sverdrup Technology, Inc. Eglin Air Force Base, Florida
ABSTRACT A computational fluid dynamic method is used to simulate relative motion between moving bodies. A CFD scheme called Beggar, is a finite volume scheme designed for the specific purpose of simulating flow about stores and store separations from aircraft. This paper describes the different aspects of the method. First, the automated grid assembly scheme, which uses polygonal mapping tree data structures is discussed. Second, the flow solver based on a Newton relaxation scheme for time-accurate flow solutions is described. A MK84 general-purpose bomb trajectory from the inboard wing station of a fighter aircraft is predicted and compared to experimental data. Also, a triple release of three generic stores from a wing is demonstrated. Key Words: Upwind, Newton, Polygonal mapping, Trajectory.
1. Introduction The USAF is continuing the development of a CFD method called Beggar. This code originated in the Wright Laboratory [BEL 95] and was transferred to the Air Force Seek Eagle Office (AFSEO) in 1993 for application. Currently, the use of CFD coupled with a dynamic rigid body solver or 6 degree-of-freedom (6-DOF), provides separation data that can be used in certification recommendations [BRK 96]. The cost per event is a fraction of a single flight test and can be repeated within 24-48 hours on multi-processing computers. Beggar also offers the advantage of easy and simplified input. All fluid boundaries such as blocked, interpolated, singularities, and freestream conditions are automatically detected. User specified boundary conditions are easily described using common English terms. Such boundaries include no-slip, tangent, and mass flow conditions among others. The solution for the coupled, time accurate, CFD and 6-DOF simulation consists of four operations: grid assembly, flow solution, force and moment calculations, and integration of the equations of motion. Each operation is executed at every time step. The flow
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calculations are run across multiple processors and executed concurrently with the other three operations [PRW 99]. 2. Grid Communication
To accommodate moving bodies, an overlapping grid system is used. Such a grid is limited only by its ability to interpolate with surrounding grids and is labeled a superblock. For relative simple geometries and those containing fixed fins, a blocked grid system (by point-to-point matching) is often desired to form a superblock. A blocked grid system can be used to construct a single superblock but no two superblocks, by definition, can construct a blocked system. Superblocks combine to form a dynamic group. Each moving body and the global static system is represented as a separate dynamic group. The core of the Automated Grid Assembly Scheme (AGAS) is the polygonal mapping (PM) tree, which is a combination of an octree and binary space partitioning (BSP) tree data structures [SAM 90]. To maximize efficiency, point classification relative to other superblocks is performed by a three-tier search. At the first level, the point is compared to a superblock bounding box to rapidly determine the general location of the point relative to a superblock. Once determined to be near a superblock, the octree data structure quickly determines the point to be in, out, or near a boundary of the superblock. Finally, if the point lies within an octant containing superblock boundary point, the BSP tree is used to determine whether the point is inside or outside the grid. 2.1 Data structure The octree data structure is one in which a region of space is recursively subdivided into octants. At n levels, a spatial domain can be divided into Sn sub-domains. A grid point can easily be located within a child octant by comparing its coordinates to that of the parent octant. The BSP data structure is one in which each node in a binary tree is represented by a plane definition. The two children of each node represent the in and out sides of a facet boundary. If a facet is clipped by an existing plane in the BSP tree, the clipped pieces of the same plane definition will be placed in different branches of the tree. The PM tree data structure makes use of both the octree and BSP data structure to increase efficiency and reduce the BSP levels. The BSP tree is used to determine whether a point is either inside or outside a grid boundary. Since it is not guaranteed to be well balanced, it is limited to the leaf octants that contains a boundary facet. The PM tree data structure is constructed by refining the octree until a single octant contains no more than one grid point from the same dynamic group. The boundary faces of the cells attached to this
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point are used to define facets inserted into the BSP tree. All superblocks in a dynamic group share a single octree data structure but the BSP tree is unique to each superblock. Once the PM tree data structure is complete, it never has to be determined again. During dynamic simulations, coordinate transformations are used to examine points that have not moved relative to the original data structure of the moving grids.
2.2. Grid assembly When a solid surface from one superblock is embedded in another superblock, a hole must be cut into the latter. AGAS uses the facets of the superblock boundary representing the solid surfaces as hole-cutters. Cells to be cut by the facets are determined from the PM tree and the nodes of the cell are labeled to be either inside or outside the cutting grid. If the facet vertices do not lie in neighboring cells then the facet is subdivided until they do. A superblock that is cut by a surface will require flow variables to be interpolated from other superblocks just outside the resulting hole. An interpolated point is one that requires its flow variables to be updated via an interpolation donor and represents a fringe or boundary point for the flow solver calculations. The interpolation donor is provide by valid fluid points from other superblocks and is determined by the PM tree. Once the PM tree identifies which superblocks the grid point that needs interpolation lies in, stencil jumping using a Newton method determines the final position of the point relative to potential donor cells. If no valid donor is found for an interpolated point, the point is labeled as out and its neighboring cells are marked as needing interpolation. The resulting hole propagates until either a valid donor is found or failure is declared.
3. Flow Solver Formulations and Schemes The Navier-Stokes equations may be written relative to a general coordinate system, £(x,y,z,t), r)(x,y,z,t} and £(x,y,z,i), in the form
where
J is the Jacobian of the inverse transformation, given by J = g^'-'sl- KjK and KUK are respectively the inviscid and viscous flux vectors associated with
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Finite volumes for complex applications
the K curvilinear coordinate. They are equal to Fj, Gj, Hj and Fv, G^, Hv if K, equals to £, 77 or £, respectively. Similarly, fj, g;, h; and fv, gv,hv are respectively the Cartesian inviscid and viscous flux vectors per unit area in the x, y and z directions, (n, v, w) is the Cartesian velocity vector, p is the density and e is the total energy per unit volume. The inviscid flux vector is given by
where The viscous terms are discretized using central differences. The thin-layer approximation which neglects cross derivative terms is used. The eigen system in the K coordinate direction is determined by the flux Jacobian AK — <9K iK /dq. This may be expressed as AK = ^A^J?"1 where RK is the matrix of right eigen vectors and AK is the diagonal matrix of eigen values. In the Steger-warming scheme, the matrix of eigen values is split into a matrix with only negative eigen values and another with only positive eigen values. This allows the splitting of the flux vector Kj K , which may be expressed as KJK = AK q, into two flux vectors associated with disturbances propagating in the negative and positive KK directions. Upwinding can then be applied separately to the two split flux vectors. In the Roe scheme, a Reimann problem (shock tube problem) is solved approximately at each time step to determine the flux vector at each of the cell faces. The initial conditions for the Reimann problem at both of the cell face sides are obtained by MUSCL extrapolation using the Van Albada limiter. The difference between the flux vector at the cell face and the flux vector associated with either of the initial conditions is expressed in terms of the summation of the flux differences across the characteristics curves propagating from the cell face. The Euler equations in characteristic form [WHI 84] along the K direction may be linearized and expressed in terms of a set of Riemann invariants. Combining the appropriate set of these equations with the far-field conditions results in the characteristic far-field boundary conditions. Noncharacteristic boundary conditions are used at solid boundaries, where a zero pressure gradient boundary condition is used in the case of inviscid flow and the no-slip boundary condition is applied in the case of viscous flow. Numerical solutions of the flow governing equations require discretized approximations of the equations. Time is discretized into finite time intervals, while the computational space is discretized into finite volumes. Discretized flow variables are assigned to each time interval and finite volume. Let i,j, k be the cell indices along the £, 77, £ coordinate directions, respectively. The discretized equations at the n+1 time step, using the first order Euler implicit time discretization, are given by
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where
provided the computational mesh is non deforming. Here, F, G and H are the numerical fluxes associated with the three curvilinear coordinates, while 6^ 6^ and 6<; are central difference operators. The nl is the index for the n+1 step. For a time accurate problem, Eq. (1) is satisfied at each time step, iteratively, by using Newton's method [WHI 91]. The application of this method leads to
where superscript m and ml are the index for the current and next Newton iteration respectively. While Eq. (1) is the discretization of the flow governing equations at a single cell, Eq. (2) combines the equations for all cells within an individual grid into a single vector equation. Here, Aq, q and R are vectors which combine the five-component vectors Aqij^,
where, superscript l is the index for the SGS iterations. Each of the iterations includes a forward pass (/z = 0) and a backward pass (n — 1). The coefficient matrix for Aq nl ' ml is split into / n l > m a block strictly lower triangular matrix, £)«.!,m a block diagonal matrix, and unl'm a block strictly upper triangular matrix. For a given inner iteration the phantom cell incremental cell solutions, Aq n l ' m l , are held fixed while the interior cell incremental solutions are
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Finite volumes for complex applications
updated. The phantom cell solutions are then updated so that the boundary conditions are satisfied. While the boundary conditions are applied explicitly for each inner iteration, at convergence of the inner iterations, an implicit updating of the boundary conditions will have been achieved [VAN 93]. 4. Dynamic Solutions
4.1. MK-84 separating from aircraft A MK-84 general-purpose bomb trajectory from the inboard wing station of an F-15E aircraft is predicted [COL 96]. The store is released in level flight at a Mach number of 0.90. The computed store trajectory is compared to experimental data obtained using the captive trajectory system (CTS) at the Arnold Engineering and Development Center's 4 foot transonic wind tunnel. The CFD solution is computed assuming inviscid flow. The total grid system consisted of 2.2 million points in 33 grids, making up 23 superblocks. Figure 1 shows the trajectory graphically through composite snapshots in time. The physical time step during the dynamic calculations was .5 milliseconds and the code was executed using a single processor on a CRAY C90. The total computer time required to complete 0.2 seconds in physical time was 55 hours. Today, this same solution can be repeated on an workstation in 24 hours. As seen in Figure 2, the trajectory compares well with experiment.
Figure 1: MK-84
Trajectory
Figure 2: MK-84 Trajectory Comparisons, Position
4.2. Ripple release demonstration For demonstrations purposes, the release of three generic finned-stores was
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621
computed [PRW 96]. This shows the versatility of the dynamic method and Figure 3 illustrates the mesh interaction between moving grids. The three stores, wing, and interface grids consisted of 2.2 million points in 16 grids (for more efficient parallelization the grids have been split to as many as 67), making up 10 superblocks, and four dynamic groups (the wing is considered a dynamic group though frozen). Figure 4 shows the trajectories graphically through composite snapshots representing 0.3 seconds in physical time from release. This configuration has been calculated on many platforms including CRAY C90 and SGI ORIGIN 2000. It is being used as a data set for research being conducted in the efficient parallelization of AGAS [PRW 99].
Figure 3: Triple Release Grid System
Figure 4: Triple Release Trajectories
5. Summary The USAF is developing and using a unique computational method for the prediction of trajectories. This method consist of an integrated set of tools that enable the classification and assemble of grids into a composite grid using novel data structures. With all grid boundaries and interpolation stencils defined, an unwind finite volume flow solver is employed to calculate fluid motion. Coupled with a 6-DOF trajectory simulation, and an efficient parallelization scheme, the capability exists to accurately and rapidly simulate aircraft and store separation events. Future development plans include multi-thread parallelization, finer grain parallelization of the AGAS, and optimization of flow solver convergence. Acknowledgements This work was performed under the direction of the SEEK EAGLE Office of the United States Air Force. The authors would like to acknowledge the
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Finite volumes for complex applications
contributing work of the entire team including Dr. Ralph Noack, Steven Standley, Shawn Westmoreland, Larry Coleman, and Joseph Keen. Special thanks to Jim Brock who is the Air Force lead for this team, Ray Maple and Nathan Prewitt for their creation and advancement of AGAS, and Dr. Davy Belk for his visionary ideas from which Beggar was created. Bibliography
[BEL 95]
BELK, D.M., MAPLE, R.C., " Automated Assembly of Structured Grids For Moving Body Problems," AIAA Paper 951680.
[BRK 96]
BROCK, JAMES, Computational Fluid Dynamics Inviscid Store Separations Verification, AFTDC TR 96-01, March 1996.
[COL 96]
COLEMAN, L.A., JOLLY, B.A., CHESSER, B.L., BROCK, J.M., "Numerical Simulation of a Store Separation Event From an F-15E Aircraft," AIAA Paper 96-3385.
[PRW96]
PREWITT, N.C, BELK, D.M., MAPLE, R.C., "Multiple Body Trajectory Calculations Using the Beggar Code," AIAA Paper 96-3384
[PRW 99]
PREWITT, N.C., BELK, D.M., SHYY, W., "Parallel Implementations of Time-Accurate CFD Applications With Moving Chimera Grids," Proceeds of the 9th SIAM Conference on Parallel Processing for Scientific Computing, March 1999.
[SAM 90]
SAMET, H.,Applications of Spatial Data Structures: Computer Graphics, Image Processing, and GIS, Addison-Wesley, 1990.
[VAN 93]
VANDEN, K.J., WHITFIELD, D.L., "Direct and Iterative Algorithms for the Three-Dimensional Euler Equations," AIAA Paper 93-3378, July 1993.
[WHI 84]
WHITFIELD, D.L., JANUS, J.M., "Three-Dimensional Unsteady Euler Equations Solution Using Flux Vector Splitting," AIAA Paper 84-1552, June 1984.
[WHI 91]
WHITFIELD, D.L., TAYLOR, L.K., "Discretized NewtonRelaxation Solution of High Resolution Flux-difference Split Schemes," AIAA Paper 91-1539, June 1991.
An Attempt to Develop a Multi Purpose FAS Multigrid Algorithm Luc Fournier*, Oliver Gloth** *INRIA, Projet Sinus, Sophia Antipolis (France) ** Institute for Combustion and Gas Dynamics, University ofDuisburg (Germany)
ABSTRACT. One of the most recent challenges in scientific computing is creating a flexible and open development platform for numerical simulations. From the implementation point of view, modern programming languages offer powerful tools for flexibility, such as the inheritance of object oriented programming or C++'s template mechanism. The numerical approach however should be flexible as well. The present paper mainly treats the numerical aspects of an FAS multigrid algorithm built in a such context to improve the numerical efficiency. To keep such a platform open for new developments, time explicit schemes together with a finite volume discretization are used. This discretization allows the usage of a volume agglomeration technique to construct a sequence of meshes. This is close to be a black box multigrid algorithm, accelerating the convergence of various problems. The main difficulties as well as approaches to overcome those are presented here. For all this parts of the FAS algorithm, flexibility in terms of the equations to be solved, as well as the discretization used, was a major guideline, although not fully achieved by now. KEYWORDS: FAS multigrid, explicit schemes, object oriented programming
1. Introduction The resolution of non linear PDEs can be done using time explicit as well as time implicit formulations. By saying explicit, it is meant that the variables on a new time level only depend on the state of the previous time level. Implicit schemes are generally based on solving a linear system of equations for every time step. They are known to be numerically efficient. Unfortunately, new physical problems or new discretizations require more implementation work. Consequently, explicit methods are suitable for an open development platform if their inefficiency for steady state problems can be overcome. Since a few decades, multigrid methods are used to accelerate the convergence properties of numerical methods. Due to the non linearity of explicit schemes, a Full Approximation Scheme (FAS) multigrid method (described in [BRA82]) has to be used for those. Explicit schemes provide an easy way of implementing different equations in an existing unstructured development platform. It seems to be natural to also find a sufficient way of generating mesh sequences for the FAS algorithm, without
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demanding too much user interaction. Please note, that user does not only stand for end users, but also for members of a research team mainly working on discretizations for physical problems. The volume agglomeration technique as presented in [LSD92] is considered as a black box method for generating finite volume coarse levels. Some more recent works have proved the efficiency of this in the field of computational fluid dynamics [MV96]. The first part of this article presents the implementation of an FAS algorithm as a basis for efficient iteration schemes for various problems. All developments presented in this paper have been done using the MOUSE C++ library [GVH97]. The development of this library has been initiated approximately two years ago. Its major intention is to be a flexible framework for finite volume computations. The second main part is devoted to some numerical experience, made with the solution of the Euler equations for compressible gas flow.
2. A multigrid approach 2.1. The FAS method The FAS multigrid method is used to solve the following non linear equation:
$(u) = S
[1]
where u denotes the field of unknowns, $ the non linear operator and S a source term. h and H, used as subscripts, refer to the fine and the coarse grid. As in every multigrid method we need inter-grid operators, iff and Iff are the operators to restrict the unknowns and the residual from the fine to the coarse grid. The prolongation of the correction from the coarse to the fine grid is written as /^. The ideal two grid FAS algorithm can be described using four steps: (k) 0
(k)
1. pre-smoothing: If u\ '' = uh ' is the initial value of the unknowns, we find a new (k) 1 estimate u\ of this field using several iterations of an explicit smoother (simply an explicit iteration scheme of the equation). rh = S^ — $h(Uh ) now represents the residual of this operation. 2. Solution of the coarse grid equation: The following equation in UH is solved.
For the ideal two grid algorithm this has to be done until a sufficient level of convergence is reached. 3. correction: Using the converged solution of [2], it is now possible to correct the variables on the fine grid.
4. post-smoothing: A new estimation uh (k) iterations of the explicit smoother, with uh
2
of the field is found, using several as initial value.
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2.2. Volume agglomeration technique The agglomeration technique is used to build a set of coarse finite volume grids. A coarser level can be obtained by clustering neighbouring volumes. Each cluster of volumes will form a control volume of the coarsened mesh. Used recursively, this technique produces a sequence of grids with nested control volumes. Figures 1 and 2 show the principle of this agglomeration algorithm applied to a mixed element sample mesh. The most used technique to agglomerate control volumes is a greedylike algorithm.
Figure 1. elements and associated control volumes
Figure 2. elements and agglomerated control volumes
But for 3D problems, Mavriplis[MV96] proposes a priority list to initialize a pure greedy technique. In [ONO98] a coloring edge-based method is found. These variants show an interest on the quality of the agglomerated cells. The basic agglomeration of the implementations presented in this paper is improved by using a local optimization criterion. It can be observed that the regularity of the agglomerated levels influences the convergence rate.
2.3. The Object Oriented Library As mentioned above the FAS method, as it is described here, is used to extend an object oriented development platform. One of it's main design principles is the encapsulation of physical, mathematical and computer related parts of the code as much as possible. The main features of this library are: - dynamic data storage classes - interpreted language for high level numerical implementations - template based operator mechanism - meshing and adaptation classes - basic visualization tool.
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The multigrid operators easily fit in this framework. Once implemented in C++ they form an extension to the control language and can thus be used to implement different iteration techniques. One major goal should be the independence of the equation to be solved. This could not be completely reached by now, since the development within this framework is fairly young. The authors however, are optimistic to be able to finish the work towards a flexible toolkit for finite volume applications using FAS for convergence acceleration. Up to date information on the developments can be obtained from http://www.vug.uni-duisburg.de/MOUSE.
2.4. Implementation of multigrid operators 2.4.1. Spatial discretization using the defect correction technique The second order flux computation is performed on the fine grid using the MUSCL interpolation [VL79]. This technique is based on a reconstruction of the state on the cell interfaces, using linear extrapolation. The extrapolation is performed using the gradients of the variables, which are computed using Green's formula for field nodes and a least square approach for boundary nodes. On the coarse levels, created by the above described agglomeration technique, the position of the nodes and the control volume borders becomes irregular. This makes a second order discretization using MUSCL extrapolation very difficult. Experiences show that such a discretization is very critical in terms of stability for the explicit time stepping. Therefore it does not seem to be reasonable to apply a higher order approach on the coarse grids. The defect correction method [DH95] proposes to solve the equation [1] iteratively by building a convergent series, based on a problem which is simple to inverse. At each iteration the following problem has to be solved: with where 3>h,i( ) and $h,2( ) respectively denote a first and a second order discretization on the fine level. Each system [4] is partially solved using one cycle of the FAS algorithm. When converged (i.e. $^1(11^ ) = $h,i(u^ )) u^ is the solution of [1].
2.4.2. Runge-Kutta for smoothing The FAS algorithm allows to treat non linear equations. On every level the exact equation, including all boundary conditions, has to be modeled. In the presented FAS implementation an explicit Runge-Kutta is used for smoothing, as well as for solving the equation on the coarsest level. For linear multigrid methods a Jacobi or GaussSeidel relaxation is often used for smoothing. In [WES91] a study of this technique for a set of linear model equations can be found. Unfortunately, this requires a point implicit formulation of the discretization. To allow the method to be easily used with different discretizations, a fully explicit Runge-Kutta method has been chosen. The
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main problem when using Runge-Kutta as a smoother, is that the smoothing properties are dependent on the number of sub-steps, the coefficients used, and the CFL number. This is shown by a von Neumann analysis of the one dimensional convection equation [5]. It is discretized using an adjustable first order upwind scheme [6]. (3 = 1 represents a full upwind and /3 = 0 a central scheme. Time integration is done by a three step Runge Kutta scheme with the coefficients o.\
Figure 3. amplification factor (J3 = ^)
Figure 4. amplification factor ((3=1)
The von Neumann analysis gives an amplification factor of this scheme as a function of the time step At and an angle 0 G [O..II]. 0 = II represents the shortest and 0 = 0 the longest resolvable wavelength. As it is noted in [WES91], the smoother does not have to be convergent. It has to be efficient for high frequency disturbances. Unfortunately, within the scope of this work, it has not been possible to apply a theoretically unstable Runge Kutta scheme with good smoothing properties for high frequencies. Figures 3 and 4 show the amplification factor as a function of At and <~). Dark colour represents a high amplification. Isolines are drawn for amplification values of (0.1,0.2,0.3,..., 1.0). Please note that values > 1 indicate an unstable scheme. 2.4.3. Boundary conditions The FAS multigrid algorithm needs a correct expression of the boundary conditions on every grid level. As explained in 2.1 a partially converged solution is obtained on each grid. On every level the same set of equations is modeled. The only difference can be found in the position of the boundary nodes. Only on the finest mesh the nodes are on the boundary itself (half cells). This makes the implementation of strong (nodal) formulations for boundary conditions easy. For the agglomerated levels this is not the case anymore. The node position can no longer be assumed to be on the boundary. Figures 5 and 6 illustrate this.
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Figure 5. Fine grid control volumes
Figure 6. 1st agglomerated level
Furthermore, geometric properties (e.g. surface normal vectors) are difficult, if not impossible, to reconstruct on the coarse levels. To still imply the boundary conditions on the coarsened levels, they are included in the source term of the defect correction scheme. The second order discretization 3>h,2(v>h ) includes a nodal formulation of the boundary conditions (see equation [4]). This formulation works well in case of the compressible Euler equations. Here a nodal boundary condition forces the velocity vectors to be parallel to surfaces. For Navier Stokes computations, where a pure Dirichlet boundary condition is demanded (non slip condition on surfaces) it is not working this way. This contradicts with the demand of a flexible algorithm. So the boundary condition problem is only partially solved by now. However, one explanation is that for the inviscid computations a weak formulation of the boundary conditions (pressure flux on surfaces) is always present. This is not the case for viscous computations. To compute a friction-caused flux on non-slip surfaces, velocity gradients are needed. Unfortunately, these gradients are very hard to compute on an agglomerated mesh. 2.4.4. Inter-grid operators Transfer operators are usually considered to be the weak part of the volume agglomeration technique. The restriction for the residual is simply a summation of all fine grid cells which contribute to one agglomerated cell. For the variable vector an averaging over all contributing cells is employed. Please note that this is done, using the control volume sizes as weighting factors. There are two different kinds of prolongation (interpolation) operators implemented. One is a canonical injection as in [LSD92], meaning that the fine grid values are simply set to the corresponding coarse grid values. Another operator is a linear interpolation using the gradients on the coarse nodes . These gradients are computed using the same combination of a Green integral and a least square algorithm as described earlier.
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3. Numerical experiments The test which is presented in this paper is a fully subsonic (Moo =0.3), inviscid and symmetric flow around a NACA0012 airfoil. Two different grids (3114 nodes and 12284 nodes) have been employed. In figure 7 the larger mesh can be seen. Figure 8 shows isolines of the Mach number for a converged solution. The inner and outer operator (3>2 and 3>i in equation [4]) are dicretized using an adjustable upwind scheme (see also equation [6]). For the inner iteration $1 a half upwind (/3 = |) has been used for all runs.
Figure 7. NACA0012 (12284 nodes)
Figure 8. Isolines of the Mach number
Convergence histories can be found in the figures 9 and 10. Each plot shows three curves (full upwind explicit, full upwind FAS, half upwind FAS). The x-axis represents the number of FAS V-cycles and the y-axis the residual. Fastest convergence could be obtained by a full upwind discretization of $2 for the defect correction scheme. The oscillatory curve in both figures represents a mono grid solution obtained by an explicit three step Runge Kutta time integration. The x-axis is scaled to represent approximately the same CPU time per unit as for the FAS solutions.
Figure 9. convergence (3114 nodes)
Figure 10. convergence (12284 nodes)
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4. Conclusions The present paper has depicted the numerical and technical aspects of a flexible (black-box like) multigrid implementation. Numerical flexibility, as it is demanded, introduces difficulties. The agglomeration technique, as well as the inter-grid operators, are not be dependent on the physical problem. However, the boundary conditions are. Thus different strategies for coarse grid boundary conditions, for different physical problems seem to be necessary. One future approach could be to classify the boundary conditions and to offer flexible and reusable solutions for every class. Dirichlet boundary conditions introduce most problems, since there are no pure boundary nodes on an agglomerated grid. If a general way of prescribing boundary values (e.g. least square extrapolation) can be formulated, the treatment of boundary conditions becomes problem independent. First numerical experiments show strong interactions between different parts of the algorithm. Therefore a better theoretical understanding of the different components (e.g. defect correction scheme, explicit smoother, inter-grid operators ...) is desired. Another challenge for this multigrid approach is the computation of time-dependent problems. A possible approach to be able to do this is a dual time stepping method [BH93]. References [BH93] M. BREUER and D. HANEL. A dual time-stepping method for 3-D, viscous, incompressible vortex flow. Comput. Fluids, N° 22, p. 467-484, 1993. [BRA82] A. BRANDT. Guide to multigrid development. Multigrid methods, in Lecture notes in Mathematics, N° 960, p. 220-312, 1982. [DH95] J.-A. DESIDERI and P. W. HEMKER. Convergence analysis of the defect-correction iteration for hyperbolic problems. SIAM J. Sci. comput., N° 16, p. 88-118, 1995. [GVH97] O. GLOTH, R. VILSMEIER, and D. HANEL. Object oriented programming for computational fluid dynamics, HiPer'97, Krakow, Poland, 1997. [LSD92] M.-H. LALLEMAND, H. STEVE, and A. DERVIEUX. Unstructured multigridding by volume agglomeration: current status. Comput. and Fluids, N°21, p.397-433, 1992. [MV96] D. J. MAVRIPLIS and V. VENKATAKRISHNAN. A 3D agglomeration multigrid solver for the reynolds-average navier-stokes equations on unstructure meshes. Int. J. for Num. Meth. in Fluids, N°23, p. 527-544, 1996. [ONO98] N. OKAMOTO, K. NAKAHASHI, and S. OBAYASHI. A coarse grid generation algorithm for agglomeration multigrid method on unstructure grids, 36th Aerospace Sciences Meeting and Exhibit, N°98-0615 in AIAA, Reno, 1998. [VL79] B. VAN LEER. Towards the ultimate conservative difference scheme V: a second-order sequel to Godunov's method. J. of Comput. Phys., N°32, p. 361-370, 1979. [WES91] P. WESSELING. An introduction to multigrid methods. John Wiley & Sons, 1991.
On Higher Order Accurate Implicit Time Advancing for Stiff Flow Problems
Cecile VIOZAT(**), Eric SCHALL(*), Alain David LESERVOISIER(*)(***)
DERVIEUX(*),
(*) INRIA, BP 93, 06902 Sophia-Antipolis Cedex (France) (**)CEA-SACLAY, 91 191 Gif-sur-Yvette cedex (France), (***)SNECMA, Centre de Villaroche, 77550 Moissy Cramayel (France)
ABSTRACT It is proposed a study of the application of a third-order accurate implicit time advancing for compressible flows at large time steps. A two-stage Runge-Kutta scheme is applied to low Mach number flows. The possibility of using an adaptive time step is addressed. Key Words: High-order time accuracy, adaptive time step, implicit RungeKutta method, unsteady flow, low Mach number, natural convection.
1. Introduction The accuracy of numerical approximation is before all determined by the amount of information devoted to the description of a given phenomenon. In advection phenomena, not only advected quantities have to be well described by the spatial mesh, but also time step size has to be fine accordingly (except for characteristic methods, for which part of the discretisation is avoided) ; the limitation on the time-step size for accuracy is expressed by the CourantFriedrichs-Lewy (CFL) condition (that is also a stability condition for explicit time advancing schemes). If we apply an implicit time advancing, using a larger time step then, either the numerical scheme is highly dissipative and dissipation errors will be dominant, or Gibbs oscillations will appear.
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Now, not only such dissipative schemes are often first order accurate ones, but also, even with high order accurate schemes, the practical accuracy cannot be better than first-order accuracy when details are not captured. By "practical accuracy order", we mean the apparent order of accuracy for a given size of time step, by opposition to the formal order of accuracy, that is an asymptotic one i.e. applying only for "small enough" time steps. This remark is in fact no true for all kind of flows. A special category is the case of low Mach flows. In the case where asymptotic models are very accurate, then their combination with implicit schemes (that are mandatory for these models) allowing highorder accuracy is a very efficient strategy (large time steps, good accuracy). In the case where a complete compressible Navier-Stokes model is applied, the question of the efficiency of a strategy relying on large time steps and high-order accurate time advancing is open. The purpose of this paper is to examine the use of a third order accurate implicit scheme in a complete compressible Navier-Stokes model with some emphasis on low Mach flows. Third-order accuracy is rarely implemented in complex CFD codes. Yet, third-order accuracy is accompanied with small phase errors for advection phenomena ; then spurious oscillations are also reduced as compared with second order accuracy. We first describe the CFD code and discusse the motivation for choosing the Norsett time advancing, then we examine the calculation of a natural convection phenomenon in which acoustics are initially predominant. The question of finding adapted time steps is finally addressed.
2. Implicit Time Advancing in CFD
We concentrate on implicit advancing for compressible flows, and more precisely on Godunov-type upwind methods. 2.1 Upwind Method In this section we describe in short the upwind spatial discretisation that we consider in the sequel. The numerical scheme is described and discussed in [DER 92]. Starting from a given triangulations, equipped with nodes attached to vertices, we build a dual control volume tesselation by dividing each triangle into three parts. A first-order accurate Godunov method for the Euler equations is written in terms of the conservative variables W = (p,pu,pv,E) by expressing the following flux equality:
where a, is the cell area, V(i) holds for the direct neighbors of vertex 2, rjjj the
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integral of normal vector along the common boundary between cell i and cell j, $(Wi, Wj, rjij) corresponds to the integration of the Euler fluxes through the common boundary with the application of a Riemann solver. Symbol "B.F." holds for boundary fluxes for which we do not want to give details here. In practice, we apply the Roe approximate Riemann solver. The right and left input of the Riemann solver are cell values Wi and Wj, which results in a standard manner into a spatially first-order (at best) scheme. In this work we use the Roe approximate Riemann solver. The necessary last feature is the (spatially) second-order accurate extension. We follow the lines of van Leer's MUSCL method ([VL 77]) and write the second order accurate fluxes in terms of cell wise reconstructions at cell interface:
In the sequel, the semi-discretised system will be written in short as follows:
where "B.T." holds for boundary terms. The above approximation is enough accurate for transonic flows, but degrades for low Mach number ones. During the last ten years, many authors have shown that the direct application of the Roe Riemann solver results in very bad approximations. In [VIO 98], some insight is given about the problems that arise in this context, and it is confirmed that the introduction of for example the Turkel preconditioning (see e.g. [VIO 98]) permits to recover accuracy for flows with low, but rather uniform Mach number. 2.2 Implicit Time Advancing : BDF1 The Backward Euler or first-order (in time) backward differencing (BDF1) scheme applied to (3) writes: where SWn+l = Wn+1 - Wn is the increment between Wn and Wn+1 corresponding to time levels n and n + 1, and in which A is the diagonal mass matrix, the input of which are:
where a}- and At" hold respectively for the area of cell i and the related time step length. When the flux function \I>2 is differentiate, it can be linearised through Taylor's formula:
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then introducing the development (5) in expression (4), we get a linearised version written under the J-form :
From a truncation analysis of scheme (6) for a ID model we derive the error order is O (At, Ax2). In practice, the flux ^2 is n°t differentiate, or difficult to differentiate, the memory requirement of the resulting sparse matrix is large and it has been found interesting to introduce linear operators that are approximations of the jacobian d^2/dW. Linearised Scheme with First Order in Space This hybrid order one/order two scheme is a simplified version of (6) where a linear operator relying on order one spatial approximation is applied to the "implicit part": where the linear operator Pn is the sum of a genuine jacobian for viscous fluxes and a first-order (in space) approximate jacobian for Euler fluxes:
Further, in the turbulent case, mean flow and closure variables are treated by separate matrices thanks to the freezing of production terms and turbulent viscosity. The advantages of this matrix is that its storage concerns only direct neighbors and it is diagonally dominant for simplified models ; this second feature allows applying rustical algorithms such as Jacobi or Gauss-Seidel. At each iteration, the linear system is actually solved by a multigrid scheme described in [FRA 98]. For unsteady laminar flows, the accuracy of the above scheme turns to be of O (At, At Ax, Ax 2 ), which in practice is not much accurate and limits the size of time steps usuable for a given accuracy. For turbulent flows, the accuracy may be degraded by the decoupling between mean variables system and closure variables system. 2.3 An Implicit Runge-Kutta Scheme We can predict that passing to second-order accuracy in time will reduce the enormous amount of dissipation that is introduced by BDF1 since in the usual truncation analysis, it is a second derivative in time that will disappear. From this point of view, stability will be notably degraded. Passing to third
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order will only "kill" a dispersion term (third order derivative), which means that the dissipation situation will essentially not change, but we reduce the phase error, an important feature for unsteady calculations. This work concentrates on passing to third-order accuracy. While second order accuracy in time is easily obtained with the extension BDF2 relying on the second-order difference in time:
that will produce an unconditionally stable scheme for both advection and diffusion, on the contrary, BDF3 is unconditionally stable only for diffusion operators. We have chosen a nonlinear scheme with two implicit systems to solve at each iteration, the Singly Diagonal two-stage Implicit Runge-Kutta (SDIRK2) scheme of Norsett [HAI 87]:
Third-order accuracy can be related by a truncation analysis to the following necessary conditions:
with 7 = (3 + \/3)/6. It is known that the above scheme is third-order accurate in time, unconditionally stable for any complex gain coefficient of nonnegative real part (A- f — J stability), which ensures stability for both advection and diffusion processes. At each stage of the SDIRK scheme, a nonlinear system has to be solved. We solve it by two nested loops. First a Newton-like or Defect Correction iteration is preconditioned by the above first-order (in space) simplified linearised system, we perform a number a of Defect Correction iterations. Second, the linearised system is solved by a multi-grid cycling. 3. A Numerical Experiment
In many transient flows of industrial interest, the Mach number is not so large, but the actual transient phenomena cover the different time scale of that kind of flow: the acoustic and convection scales. For representing this kind of
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Figure 1. Various time steps length Figure 3. Nusselt time number for calculations started from time with different strategies. t = 0.1. a =5, Ra= 103.
obtained
flow, we consider a transient version of the test case of De Vahl Davis [DVD 83] by specifying an initial condition : the fluid in the square cavity is at rest with uniform temperature equal to the (right horizontal) cold wall. Evolution will be forced by putting suddenly the temperature of the left vertical wall to a larger level and maintaining it at this level, the warm wall temperature. After fast acoustic phenomena, convection will install and progressively get steady. The Mach number in this case is of the order of 10- 5 . 3.1 A Global Computation Since time scale varies a lot, a constant time step is not adapted to the whole calculation. After some trial and errors we have succeeded in computing the flow with some representation of each phase by imposing the time step to increase linearly with the physical time. For result evaluation we concentrated on the Nusselt output, a number indicating the rate of calories through the cold wall. For presenting its evolution as a function of time we have to use a logarithmic scale for time (Figure 3). We concentrate on the final phase of the flow. In Figure 1, we observe that for a much larger time step, the third-order time advancing is still much more accurate; in fact, for a same level of accuracy, SDIRK is 16 times less expensive. 3.2 Adaptive Time Stepping Since both time advancing schemes are a priori unconditionally stable, time steps as large as we want can be used stably. But accuracy can be a mess. Indeed, for BDF1 applications, waves which are smaller than time discretisation may be completely damped, and for SDIRK applications, too large time steps may produce results much less good than BDF1. This is why we have to
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Figure 2. Time step adaptation : time step length for a sensor based on ei(t) (left) and for a sensor based on CAvg(t) (right).
consider the automatic adaptation of the time step length. Such adaptation is not so easy due to a particular characteristic of the flow: acoustic waves, that are quasi periodic, and that present amplitudes decreasing progressively of several order of magnitude. Many classical choice in measuring the time truncation will be faced to the resulting signal, with large oscillations, leading to too large and too small time steps. For example, a sensor relying on maximum (in domain) of the pressure time derivative:
will produce the "thick" signal presented in Figure 2 (on the left). We consider and its average where nt is the number of nodes. The actual sensor that we propose is: const. The ability of this sensor in evaluating the heterogeneity of the pressure field is much more stable, since, taking the inverse as a time step produces a rather regular time step as illustrated in Figure 2 (on the right); in fact this adaptive time step turns to be very similar to the one obtained by trial and errors. In Figure 3, an adaptive computation with a reinforced iterative convergence at each stage is shown to produce a better result (with larger time steps) than a previous version especially for the final phase of the flow.
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4. Conclusion
We have shown that third order time accuracy can be observed on nontrivial problems computed with implicit time advancing. The explanation is that pressure fluctuations, that are too fast for a good representation with a large time step are in fact of too small amplitude for interfering with the overall accuracy measure. From this point of view, full compressible approximations enjoy the same advantage as asymptotical models. For too large time steps, however, we emphasize that the third order scheme can be less accurate than the first order BDF1. This is why such schemes have to be applied in combination with adapted or adaptive time step length. We have addressed the problem of choosing such an adaptive time step for a rather particular class of flows in which pronounced periodicity was a serious obstacle to a standard timetruncation evaluation. Although still a prospective choice, it proved to be an efficient solution for the problem under investigation.
Acknowledgement s
The authors wish to thank Roger Peyret and Jerome Francescatto for their kind help in this study.
Bibliography
[DER 92]
A. DERVIEUX, J. A. DESIDERI, Compressible Flow Solvers using Unstructured Grid, Rapport INRIA, No 1732, 1992.
[HAI 87]
E. HAIRER, S.P. NORSTETT, G. WANNER., Solving Ordinary Differential Equations I, Springer-Verlag, 1987.
[VL 77]
B. VAN LEER, Upstream Centered Finite Difference Schemes for Ideal Compressible Flow, J.Comp.Phys., Vol. 23, pp. 263275, 1977.
[VIO 98]
C. VIOZAT, "Ecoulements stationnaires et instationnaires a petit nombre de Mach et maillages etires", These a I'universite de Nice, 1998.
[FRA 98]
J. FRANCESCATTO, A. DERVIEUX,"A semi-coarsening strategy for unstructured MG based on agglomeration", Int. J. Num. Meth. Fluids, 26, pp. 927-957, 1998.
[DVD 83]
G. DE VAHL DAVIS,"Natural convection of air in a square cavity : a benchmark numerical solution", Int. J. Numer. Methods Fluids, Vol. 3, pp. 227-248, 1983.
Numerical Solution of Steady 2D and 3D Impinging Jet Flows
Karel Kozel, Petr Louda, Jaromir Pfihoda
Department of Technical Mathematics CTU Prague Karlovo ndmesti 13 CZ-121 35 Praha 2 Institute of Thermomechanics AS CR Dolejskova 5 CZ-182 00 Praha 8
ABSTRACT This paper deals with the numerical solution of 2D and 3D incompressible turbulent impinging jet flows. Numerical method is based on an artificial compressibility method, discretisation is done using cell-centered finite volume method with quadrilateral and hexahedral cells in 2D and 3D case respectively, applied to structured multi-block grid. We used several explicit and implicit schemes and tested their efficiency. The performance of eddy viscosity low-Re two-equation turbulence models is tested. Modification of Chien k-c model is proposed, which excludes the direct dependence on friction velocity. In 2D case, an algebraic Reynolds stress model is also used. Key Words: artificial compressibility method, explicit and implicit schemes, incompressible impinging jet, turbulence models
1. Introduction The subject of this study is to compare predictions of the averaged flow field in the steady incompressible impinging jet provided by the different turbulence
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models within the framework of artificial compressibility finite volume method. The turbulence models considered are low Reynolds number eddy-viscosity two-equation models by Launder-Sharma, Chien (see [PAT 85]), and YangShih k-e, and Menter's SST model ([MEN 94]). The Chien's version, however, is modified in order not to depend explicitly on friction velocity. For 2D mean flow, a non-linear model of Reynolds stress ([PAT 99]), recently developed at the Dept. of Mechanical and Thermal Engineering, RUG Gent, is being tested. This model implicitly includes the influence of streamline curvature. Since it requires the kinetic energy and time scale of turbulence to be specified, it is used in conjunction with linear eddy viscosity models solving for these quantities. 2. Mathematical model The turbulent flow of incompressible newtonian fluid in the volume V is supposed to satisfy Reynolds-averaged Navier-Stokes equations in the form
where Vi denotes (time) averaged velocity vector, t time, Ui unit outer normal vector of the surface <9V, aij averaged stress tensor given by where p is averaged static pressure divided by constant density of fluid, <5^ Kronecker's delta, V kinematic viscosity, Sij = \ (|^- + ^-J strain rate tensor, and Tij is Reynolds stress tensor. 2.1 Turbulence model To solve equation (1) for unknown velocity and pressure, the Reynolds stress tensor must be specified. The linear model follows Bousinesq's hypothesis, the non-linear one contains terms of second and third order in terms of power of mean velocity derivatives; the third order term captures the effect of rotation. The constitutive relation for deviatoric part of Reynolds stresses in linear model is given by
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For 2D mean flow, the general representation of Reynolds stresses is as follows
where strain rate and rotation tensors are non-dimensionalised by the time scale of turbulence r and their invariants are
and C"s are functions of invariants S and Cl. When using constitutive relations for Reynolds stress mentioned above, one has to specify A;, VT, and r. In this study, the low-Reynolds number twoequation models are used. The generic form of modelled equations for turbulence kinetic energy and dissipation rate is then
where q stands for k or e, and eddy viscosity is given by
Considering the modified Chien k-e model, the source terms are given by
where (with y being the distance to the solid wall)
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3. Numerical solution In order to obtain the stationary solution W = col(p,u,v,w,k,e) of the system of equations (1) and (2)
with steady boundary conditions, we use the artificial compressibility method, i.e. we seek solution of the above system now with Q = diag(1,1,1,1,1,1) and the same boundary conditions for t —>• oo. 3.1 Discretization in space To discretize equation (3) in space, the cell-centered finite-volume method on a structured grid consisting of quadrilateral or hexahedral cells in 2D or 3D case respectively is used. Approximating convective transport of the quantity q, we use AUSM type scheme ([VIE 98]), which for hexahedral cell i in structured grid reads
where the unit flux
through the cell face / between cells i and i + 1 is and value qi is interpolated by
with e.g. K — 1 for central scheme and K = 1/3 for 3-rd order upwind scheme. Discretization of non-viscous stresses is central, as well as for viscous ones. Derivatives in viscous stresses are approximated by their mean values in the dual octahedral cell constructed over each face / with vertices located on vertices of face / and in centres of the cells i and i + I. For stability to be maintained, the second order differences based artificial damping of pressure is added to the right hand side of continuity equation. 3.1 Discretization in time After dicretization in space is done, the system to solve becomes
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with W being the cell average value. We use m-stage Runge-Kutta scheme, which has the form
For e.g. 3-stage scheme is scheme Following scheme is the implicit one
, and for 4-stage
with Newton linearization of non-linear terms in the cell i:
(Hi is set of cells in the stencil) we obtain linear algebraic system with 13 or 33-block-diagonal matrix in 2D or 3D case respectively. The system is solved iteratively using block tri-diagonal systems in one direction and relaxation in other directions. Only the negative part of source terms in turbulence models is treated implicitly, with approximate, empirical, Jacobi matrices. The first and third order terms of Reynolds stress is treated implicitly as long as the effective viscosity is positive (invariants S, £1 are kept constant during time step), otherwise explicitly as well as the second order term. 4. Numerical results
The grid for 2D cases is orthogonal. Fig.l compares the efficiency (in terms of number of time steps required) of the central 3-stage Runge-Kutta scheme and implicit scheme (a — 1/2, CFL — 50) with other 2 schemes not described here (see [KOZ 97]) using log Z/2 norm of steady residuals. Fig.3 shows multiblock grid in (z,y)-plane for 3D jet with axisymetric geometry. In z-direction we consider both confined and semi-confined geometry (ERCOFTAC's testcase) - there is only small difference in static pressure on the impingement wall between both cases for the distance of the walls equal 2 diameters of inlet pipe (fig.2). Isolines of k in 3D case, by Launder-Sharma k-e model and implicit scheme (a = 1, CFL = 400), are shown on fig.6. Figures 4 and 5 show velocity and turbulent kinetic energy k on the jet axis of 2D impinging jet. Although
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Figure 1: 2D laminar impinging jet (Re = 50) - convergence histories non-linear model of Reynolds stress gives higher k around stagnation point, the mean velocity is almost the same as achieved by linear model. Figure 8 shows velocity vectors of 3D confined impinging jet.
Fig. 2: 3D turbulent impinging jet (Re = 28000J - static pressure along impingement wall
Fig. 3: Grid in the (x,y)-plane for 3D jet
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Fig. 4: 2D turbulent impinging jet (Re - 20000J - velocity on the jet axis
Fig. 6: 3D turbulent impinging jet (Re — 28000,) - isolines of k and grid used
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Fig. 5: 2D turbulent impinging jet (Re = 20000,) - turbulent kinetic energy on the jet axis
Fig. 7: Isolines of turbulent kinetic energy in 2D impinging jet (non-linear model, Re — 12000,)
Figure 8: Velocity vectors for confined 3D impinging jet (Re — 20000).
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Bibliography
[PAT 85]
PATEL V. C., RODI W., SCHEUERER G.: Turbulence Models for Near-Wall and Low Reynolds Number Flows: A Review, AIAA Journal, Vol.23, 1985, pp. 1308-1319
[MEN 94]
MENTER F. R.: Two-Equations Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA Journal, Vol.32, No.8, 1994, pp. 1598-1605
[KOZ 97]
KOZEL K., LOUDA P.: Numerical Solution of 2D and 3D Free and Impinging Jet, Proceedings of Conference "Numerical Modelling in Continuum Mechanics", Part II, Prague 1997, Ed. M.Feistauer, R.Rannacher and K.Kozel, pp. 367-374
[VIE 98]
VIERENDEELS J., RiEMSLAGH K., DICK E.: A Multigrid Semi-Implicit Line-Method fo Viscous Incompressible Flows, Proceedings of Seminar "Euler and Navier-Stokes Equations", Prague 1998, Ed. K.Kozel, J.Pnhoda, M.Feistauer, pp. 89-92
[PAT 99]
PATTUN S.: Non-linear, Low-Reynolds, Two-Equation Turbulence Models, Ph.D. Thesis, RUG Gent, 1999 (http://allserv.rug.ac.be/~edick/tw03v/saskia/ index.html)
Acknowledgement This work was partially supported by grants No.lOl/98/KOOl of GA CR, COST OC Fl.70/97 and Research Project of MSMT No.304/98/210000010.
Triangular, Dual and Barycentric Finite Volumes in Fluid Dynamics* Jifi Felcman and Miloslav Feistauer Institute of Numerical Mathematics Faculty of Mathematics and Physics Charles University, Prague Malostranske nam. 25 HSOOPrahal Czech Republic
"The present research has been supported under the grant No. 201/99/0267 of the Czech Grant Agency and the Grant No. CEZ J13/98,113200007.
ABSTRACT The robust adaptive higher order method is devised for the numerical simulation of the compressible gas flow. Since the viscosity and heat conduction coefficients of gases are small, the viscous dissipative terms could be considered as perturbations of the inviscid Euler system. Therefore the method is based on general class of flux vector splitting schemes for the finite volume solution of the inviscid Euler equations. The structure of the viscous terms offers the application of the finite element method. The choice of the shape of finite volumes with respect to the use of conforming/nonconforming finite elements is demonstrated on practical examples. Key Words: Navier-Stokes equations, transonic flow, finite volume method.
1. Formulation of the problem
We consider gas flow in a space-time cylinder QT = 1) x (0,T), where fJ C ./R2 is a bounded polygonal domain representing the region occupied by the fluid and T > 0. The complete system of viscous compressible flow consisting of the continuity equation, Navier-Stokes equations and energy equation can be written in the form
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for the unknown vector-valued function of conserved quantities w. The functions fs and Rs are called inviscid (Euler) and viscous fluxes, respectively. The above system is equipped with initial and boundary conditions. For their definition including the periodicity condition in the solution of a cascade now see [FFL 95].
2. Discretization We split (1) into the inviscid system and the purely viscous system (2)
(3)
and discretize them separately. The time discretization of (2) and (3) is carried out with the use of a partition 0 = to < ti < £2 < • • • of the time interval (0, T) by the forward Euler method. The inviscid system (2) is discretized by the cell-centred finite volume (FV) method on a mesh T>h — {Dj}j £ / (/ is a suitable index set), where the finite volumes Di are closed polygons. The boundary dDi can be expressed in the form dDi — Uigsfi) ^iji where F^- is either the intersection dDi D dDj or FJJ C <90. S(i} is a suitable index set. We set \Di\ = area of Di, n^- = unit outer normal to dDi on F^, hij = length of F^. The averages of the sought solution w on Di at time tk are approximated by values to*. The purely viscous system (3) is discretized by the finite element (FE) method on a mesh 7~h, that generally differs from T>h used for the finite volume method. Using the above ideas, we discretize the complete system (1) via operator inviscid-viscous splitting. One time step tk —> tk+i is divided into two fractional steps. Step I (inviscid FV step on the mesh D/J: Assume that the values to*, i 6 /, approximating the solution on the finite volumes Di at time tk are known. k+Compute the values to, 2, i G /, from the FV formula
equipped with inviscid boundary conditions. In (4), H is a suitable numerical flux. For its construction see e.g. [FEI 93]. From the point of the adaptive mesh refinement the Osher - Solomon numerical flux seems to be the most
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robust (for the numerical tests see [FDF 94, FFD 96]). The higher order extension of the scheme (4) was proposed in [DFE 98]. For a 3D version see [FSO 98]. Step II (viscous FE step on the mesh Th)'- define the finite element function &-)_! fcjI wh 2 from the values wi 2 computed in the FV step. Apply the viscous Dirichlet boundary conditions. Compute the finite element function wh+1 as the solution of the following problem satisfies the viscous Dirichlet conditions, ,
for all test functions tf>h 6 Vh- The definition of Vh corresponds to the choice of the used finite volume mesh T>h and the type of Dirichlet boundary conditions in the viscous problem. In (5), (w, ip)h and a,h(w, <£>) denote the approximation of Jn w (f> dx and /n Y^s=i ^-s(w,(f>) -^2-dx, respectively, obtained with the aid of the numerical quadrature. The above system (4)-(5) is purely explicit. The stability condition from [FFD 96] is used.
3. Combined FV-FE method Often the structured quadrilateral FV grids are used in the numerical solution of the Euler equations. Despite their good resolution of channel flow (e. g. GAMM channel) and flow arround an airfoil (e. g. NACA 0012) the results for cascade flow (due to the complicated domain and periodic boundary conditions) are rather poor (cf. [FDF 94]). Therefore we abandoned them and concentrate us on triangular generally unstructed meshes. We denote the primary triangular mesh in the combined FV-FE method by <Sh — {Tj}jej (see Fig. 1). From this mesh the unstructured FV grid "Z\ and FE grid Th are derived. 3.1 Dual FV and triangular FE Denoting by ah = {Pj}je/ the set of all vertices of all triangles T e Sh, we associate each Pj € ah with the dual finite volume Di constructed in the following way: Join the centre of gravity of every triangle T € <Sh, containing the vertex Pj, with the centre of every side of T containing Pi. If Pi € ah f) <9fl, then we complete the obtained contour by the straight segments joining Pj with the centres of boundary sides that contain Pj. In this way we get the boundary dDi of the finite volume Di. (See Fig. 2.) We set Dh '•=• {-Di}ie/ and Th '•= $hWe use conforming piecewise linear finite elements for the discretization of purely viscous system (3). The components of the vector w are approximated
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by functions from the finite dimensional space is linear for each T We set in (5)
on the part of where Wi satisfies the Dirichlet condition L,
I i
L,
1
i
JUI
i
and define wh 2 G Xh such that -2;^ 2 (Pi) = w{ 2, i £ /. The numerical quadrature, called mass lumping, is used in the evaluation of (-, •)& and a/ l (-, •) in (5) (see [FFL 95]). 3.2 Triangular FV and triangular FE The meshT>h = <Sh is used in the discrete scheme (4) (see Fig. 3). The conforming linear finite elements on a triangulation Th of il are used in the scheme (5). The triangulation Th is compatible with T>h in the following sense: The set of all vertices of the triangles T £ Th consists of the barycentres Bi, i 6 /, of all Di £ T>h and the vertices of D{ 6 T>h lying on <9f7. In this case the spaces Xh and V/i are defined in the same way as in (6) and (7). 3.3 Barycentric FV and triangular FE By {Si}i£j we denote the set of all sides of all T e 5^. We construct the barycenter finite volume mesh T>h — {Di}i^i over the grid Th- Each side Si <£. dft is associated with the quadrilateral Di for which the boundary consists of segments connecting the endpoints of Si with the barycentres of the triangles adjacent to the side Si. For 5^ C <9fJ, Di is defined as the triangle with sides formed by the above segments and Si. (See Fig. 4.) The finite volume approximation of the exact solution w at time tk is sought as a piecewise constant function in fJ with constant values on Di, i £ /. The scheme (4) is applied on the mesh T>h • The finite element approximation (5) uses the Crouzeix - Raviart nonconforming finite elements (cf. [FEI 93, Par. 8.9]), linear on each T £ Th '•= <$h and continuous at midpoints of 5$, i € /. The components of the vector w are approximated by functions from the space is linear for each T £ Th, y is continuous at midpoint of Si for each at the midpoint of
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where Wi satisfies the Dirichlet condition. > (9) We define wkh+'2 e X%R such that #J + '(Qi) = w*+*, where Q» is the midpoint of the side Si G (Sjjje/- The numerical quadrature using the midpoints ~ k+2
as integration points is used in (5). The scheme (5) with wh and the test function space V^R is applied.
defined above
4. Computational results The combined FV-FE method was applied to several test problems. The goal was to obtain steady viscous transonic flow via time stabilization for tk —> oo. Here we present the application of this method to a technically relevant flow past a cascade of profiles. The results were obtained in cooperation with V. Dolejsi. The computational results are compared with a wind tunnel experiment (by courtesy of the Institute of Thermomechanics of the Czech Academy of Sciences in Prague). The experiment and the computations were performed for the following data: angle of attack = 19° 18', inlet Mach number = 0.32, outlet Mach number = 1.18, 7 = 1.4, Reynolds number w 1.5 • 106. Two cases, namely triangular FV/conforming FE and barycentricFV/nonconformingFE were computed. Fig. 5 and 6 show the meshl^ and Th for both cases. The initial coarse triangular mesh was refined adaptively according to the algorithm described in [DOL 98]. Fig. 5 and Fig. 6, where the pressure distribution along the profile is plotted, indicate a very good agreement of computational (—) and experimental (<0>) results for the case from Fig. 9 and Fig. 10, respectively. In order to support theoretically the computational results, the combined FV - FE schemes were analyzed in the case of a model scalar nonlinear convectiondiffusion problem. In [FFL 97] and [FSS 99] the convergence of the method combining dual FV and conforming FE is proven for semiimplicit and purely explicit schemes, respectively. The work [FLW 99] is concerned with error estimates of the semiimplicit method. In a similar, but more sophisticated way the combined barycentric FV/triangular FE method was analysed in [ADF 98] and [DFK 99]. References [ADF 98]
Angot Ph., Dolejsi V., Feistauer M., Felcman J., Analysis of a combined barycentric finite volume - finite element method for nonlinear convection-diffusion problems, Appl. Math. 43, No. 4, (1998), 263-310.
[DOL 98]
Dolejsi V., Sur des methodes combinant des volumes finis et des elements finis pour le calcul d'ecoulements compressibles sur des
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Finite volumes for complex applications maillages non structures, PhD disertation, Charles University & Universite Mediterranee, 1998.
[DFK 99]
Dolejsi V., Feistauer M., Klikova A., Error estimates for bary-centric finite volumes combined with nonconforming finite elements applied to nonlinear convection-diffusion problems. Preprint. Faculty of Mathematics and Physics, Praha, 1999.
[DFE 98]
Dolejsi V., Felcman J., Investigation of order of accuracy for higher order finite volume schemes, Engineering Mechanics, Vol. 5, No. 5, (1998), 327-337.
[FEI 93]
Feistauer M., Mathematical Methods in Fluid Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 67 (Longman Scientific & Technical, Harlow, 1993).
[FFD 96]
Feistauer M., Felcman J., Dolejsi V., Numerical simulation of compressible viscous fiow through cascades of profiles, ZAMM 76 (1996) S4, 297-300.
[FFL 95]
Feistauer M., Felcman J., Lukacova-Medvid'ova M., Combined finite element - finite volume solution of compressible fiow, J. Comput. Appl. Math. 63 (1995), 179-199.
[FFL 97]
Feistauer M., Felcman J., Lukacova-Medvid'ova M., On the convergence of a combined finite-volume - finite element method for nonlinear convection-diffusion problems, Numer. Methods Partial Differential Eq. 13 (1997), 163-190.
[FLW 99]
Feistauer M., Felcman J., Lukacova-Medvid'ova M., Warnecke G., Error estimates of a combined finite volume - finite element method for nonlinear convection-diffusion problems, accepted in SIAM J. Num. Anal.
[FSS 99]
Feistauer M., Slavik J., Stupka P., On the convergence of a combined finite volume - finite element method for nonlinear convectiondiffusion problems. Explicit schemes, Numer. Methods Partial Differential Eq., 15 (1999), 215-235.
[FDO 96]
Felcman J., Dolejsi V., Adaptive methods for the solution of the Euler equations in elements of blade machines, ZAMM 76 (1996), S4, 301-304.
[FDF 94]
Felcman J., Dolejsi V., Feistauer M., Adaptive finite volume method for the numerical solution of the compressible Euler equations, Proceedingsof the Second European Computational Fluid Dynamics Conference, ECCOMAS94, 5-8 September 1994, Stuttgart, Germany, John Wiley and Sons, 894-901, 1994.
[FSO 98]
Felcman J., Solin P., On the construction of the Osher-Solomon scheme for the 3D Euler equations, East-West J. Numer. Math., Vol. 6, No. 1 (1998).
Complexity, performance and informatics
Figure 1. Triangular mesh Sh
Figure 2. Dual FV mesh T>h and triangular FE mesh Th
Figure 3. Triangular FV mesh T>h and triangular FE mesh Th
Figure 4. Barycentric FV mesh T>h and triangular FE mesh Th
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Figure 5. Triangular FV and conforming FE - pressure distribution
0
0.2
0.4
0.6
0.8
1
Figure 6. Barycentric FV and nonconforming FE - pressure distribution
Concepts for parallel numerical solution of PDEs
Guntram Berti Institut fur Mathematik, BTU Cottbus Universitdtsplatz 3-4 03044 Cottbus, Germany bertiOmath.tu-cottbus.de
ABSTRACT The parallelization of numerical codes on unstructured grids still is a demanding programming task, which is often performed in an application-specific manner. However, many of the algorithms used in this field exhibit strong structural similarities. A concept exploiting the latter is developed in this paper. Furthermore, it will be shown how the abstract concepts can be implemented in a reusable generic way, thus easing the actual work to be done for building parallel PDE software. Keywords: parallel scientific computing, distributed grid data structures
1. Introduction For applications in the field of numerical PDE solution, the increased computational power of parallel hardware offers a promising way of surpassing limitations of resources. However, parallelization of numerical code can be a demanding task, especially when performed from scratch. Taking advantage of existing tools for parallel computing is therefore a sensible option. For applications operating on regularly structured data (e. g. Cartesian grids) a good deal of tools are available. For unstructured grids, on the other hand, there is still a lack of sufficient tool support, see [Bir98] for a different approach and an overview. We propose a generic, data-structure based solution, which makes only minor modifications to existing code necessary. This is no automatic parallelization concept; the act of data exchange remains under the control of the application program.
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Finite volumes for complex applications
Our approach emphasizes logical grid distribution; it is applicable for distributed as well as for shared memory, and for multi-block type grids. The organization of the paper is as follows: First, a short overview over the strategy of geometric partitioning is given. Then the concepts of distributed overlapping grids are introduced, and some aspects of their implementation are discussed. Finally, we review some implementation-related issues.
2. Geometric Partitioning for parallel PDE solution Algorithms for grid-based numerical PDE solution typically work locally on data with a predictable pattern of access, with work roughly proportional to grid size. Many of these algorithms do not depend on the order grid elements are processed (this is not true e. g. for some linear solvers used for convectiondominated problems). Therefore, geometric partitioning, or domain decomposition, is an adequate parallelization strategy [Zom96]: Equal-sized parts of the computational grid are assigned to different processes. These parts have an overlap which acts like a cache in order to avoid direct access to remote data; its size depends on the client algorithms used by the application. Computations in parallel (ownercomputes rule) alternate with synchronizing the locally copied information in the overlap with the neighbor processes. In order to implement domain decomposition, several concrete problems have to be solved. A given grid has to be partitioned into equal parts; an appropriate overlap has to be derived from the algorithms and to be added to these parts. We must maintain some correspondence relation between grid regions duplicated in the overlap, and provide a means to ensure consistency of duplicated data on the grid. These question have to be tackled in the basic case of static distribution. If the grid distribution is dynamic, there are the additional difficulties of incremental load re-balancing and migration of grid-related data structures. We concentrate here on the static case, because it is required for every application. Also, it covers most of the parallel application-level code as well as the actual communications load, so it is the key to both parallel performance and ease-of-use. Migrating a distributed grid data structure definitely is not a trivial task, but can be made largely transparent to the application-level code. This problem will be discussed in [Ber99j. We do not treat partitioning of grids in this paper. It is a well-investigated problem ([Els97]) and is considered a preprocessing step here. 3. Concepts for distributed grids Manageable data structures for distributed grids are the key to parallel implementations of PDE solvers. They should both hide any detail of no interest
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to the application programmer and at the same time provide the high-level constructs needed to express the logical nature of data distribution. In particular, a simple yet expressive way of dealing with the logical parts of grid overlap according to their ownership semantics proves essential. The following three sections are concerned respectively with the inner structure of the overlap, maintaining consistency of data on the grid (distributed grid functions), and finally the construction of the overlap itself, driven by the stencils of client algorithms. 3.1.
Distributed overlapping grids — DOGs
Our point of departure is a simple 'sequential' grid, henceforth called local grid. For parallelization of existing code, it is crucial that the representation of this grid does not need to be changed. On top of this local grid sits an overlapping grid, which manages overlap-related information. It also has access to a coarse partition grid, the cells of which correspond to the local grids, and establishes the link to neighboring local grids.
Figure 1: components of a distributed grid Thus far, the grid entities have a local meaning only. Now, we add as an additional layer a distributed grid, which stands for the ensemble of all local grids. If there is single global memory, there needs to be only one such entity that contains the array of local overlapping grids. In the case of a physically distributed environment, this layer is represented by a distributed grid for each physical memory. The distinction between overlapping and distributed grid is essential for the meaning of global reduction operations. It also leads to a better factorization of functionality, such that an overlapping grid implementation can be reused for many distribution scenarios.
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Finite volumes for complex applications
The concepts of overlap structure will be introduced here in an informal manner; the reader interested in a formal treatment is referred to [Ber99]. As mentioned above, each local grid Qi contains a certain overlap Oi which can be thought of as a 'halo' surrounding it. The local grid parts are tied together by means of identification mappings between their overlaps, defining the global grid Q as the union of the Qi with the proper identifications. On each local overlap, we impose ownership relations by distinguishing between parts £ij that belong to Qi (and are copied in Qj], ranges Sij that are shared between Qi and Qj, and parts Cij that are copied from Qj, cf. fig. 2. These bilateral ranges support the efficient data-exchange among neighbors. They are complemented by total ranges Ei,Si and C{ given by union over all neighbors j. These are central for guiding the parallel execution of algorithms. For completeness sake, we mention also private ranges Ti not duplicated elsewhere, imported ranges (Si U Ci) and exported ranges £i U <Sj. 3.2. Distributed grid functions - DGFs Static communication patterns exchange only data on the grid, a notion captured by the concept of distributed grid functions. As before, we distinguish between local ('sequential'), overlapping, distributed and global grid functions, corresponding to the type of grid they are thought to 'live on'. For a proper definition of the notions of consistency, an overlapping grid function F has to be considered jointly with an algorithm # responsible for calculating its values, a write-range W where F is locally computed, and a read-range R, where F is locally accessed. What about the class of algorithms # we consider here? First, according to the SPMD paradigm, £ is thought to be a function on the global grid computing a global grid function F. This can only be achieved by computing locally the Fi if # is data-parallel: It does not depend on the order of computation, i. e. not on the values of F itself. If this not the case (e. g. Gauss-Seidel iteration), we obtain another algorithm $p which is a block-version of the original #. This may or may not be tolerable, and must be decided by the numerical scientist. Assuming now # is 'sufficiently data-parallel', we call a tuple (F,$,W,R) locally up-to-date if F(e) = $(e) for all e e W; it is locally consistent if also F(e) = 3"(e) for all e 6 R with remotely owned or shared counterparts e. A distributed grid function (Fi,3;,Wi,Ri)i
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Second, we have to control when a synchronization should occur. Again, this is safe when all local computations are done; we can overlap computation and communication (compute-and-send-ahead) by first calculating on exported ranges, initializing data exchange, calculating the rest, and finishing data exchange. The task performed by DGFs may be a bit more involved. So it is sometimes useful to calculate results only partially on the shared parts of W', e. g. the assemblation of FEM stiffness matrices. Then synchronization means reduction (typically addition) on shared ranges. It may also be useful to work with data that is only locally up-to-date, and it depends on the sequence of algebraic operations, when a consistent state has to be established, see [Bas94]. 3.3.
Incidence-based stencils and overlap generation
The efficiency of the approach strongly depends on the size of the local support or stencil of #, which is for each grid element e the minimal neighborhood U(e) such that the evaluation of $(e) uses only information defined on U(e). Two questions now arise: how can the stencils of algorithms on unstructured grids be described, and how can one produce the correct overlap from that and a partitioned grid? Attacking the first problem, we confine ourselves to stencils expressible in terms of the grids incidence relations. To formalize this idea, we introduce incidence sequences I — (do, • • • , dn) with 0 < di < d = dim(^), meaning "go from elements of dim. do to incident elements of dim. d\ to ... ". Given a set A (or germ) of elements of dimension do, we may define the hull l-Lj(A} generated by / on A by taking the union of all elements visited by following the path given by /, starting from the elements of A.
(a) stencil (d, d— l,d)
(b) corresponding hulls
Figure 2: Simple stencil and overlap generated by MARK-HULL
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Finite volumes for complex applications
(a) stencil (d, l , d )
(b) corresponding hulls
Figure 3: Another stencil and overlap generated for one grid part
Consider the following simple flux calculation example: for all Cells c e Q do for all Cells n adjacent to c do fluxsum(c) 4-fluxsum(c) + flux(U(c),U(n))
The stencil of this algorithm is given by (d, d — l,d), which means that for a given cell, each neighbor cell is accessed, via the corresponding facet, as depicted by figure 2(a). If some kind of second-order recovery is added, involving interpolation on vertices, a more complicated stencil results (figure 3). Knowing this local stencil, it is now possible to derive an appropriate overlap, given a partitioned grid. In essence, this reduces to the calculation of the hull generated by the stencil, taking the boundary of the grid partitions as germs. An algorithm MARK-HULL using space and time proportional to the size of the generated hull has been developed and implemented, but cannot be reproduced here due to space limitations, see [Ber99] for a discussion. For the flux-algorithm stencil above, this results in an overlap as infigure2(b). In a typical application, there are several stencils involved, and thus different hulls may need to be computed. The total overlap of the grid is then composed by the union of partial overlaps imposed by different algorithms. 4. Generic implementation and practical experience
The concepts presented so far are useful in their own right. However, we would like to offer an implementation which is usable without change for a large variety of concrete grid data structures. In order to achieve such a grid-independent realization, it proves necessary to adopt a more abstract approach to grid data structures. In [BB99] is shown how an abstract grid interface enables a generic programming style, producing implementations that do not rely on a particular
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grid representation. See [Ber99] for a detailed description of these concepts. We just remark that our approach makes a clear separation between combinatorial grid, geometric embedding of the grid and grid functions that allow to store data on grid elements. This corresponds well to the requirements of the distributed grid data structures, as it mainly depends on the combinatorial properties, and in the case of a static distribution, grid functions are the main actors. Most importantly, incidence information is captured by iterators, which essentially answer queries like "give me all vertices incident to this cell, one after another", making them a suitable basis for the MARK-HULL algorithm mentioned above. On top of this grid abstraction, data structures for representing distributed grids, overlaps and grid ranges, as well as the algorithms for generating them from incidence-type stencils, have been developed in a fully generic fashion. Thus, the implementation may be directly used with any grid that satisfies the abstract interface. In practice, it proves rather straightforward to adapt existing grid implementations to this interface, or more precisely, to provide a separate implementation for it. According to our experiences, this requires only one or two days of work, which could probably still further reduced. Once this has been done, the generic implementation of distributed grids can be used 'off the shelf, fig. 4. It remains to insert the appropriate calls to synchronization routines and global reduction operations. Most of the code Figure 4= using generic components with exmay continue to use the 'native' grid isting codes data structures. It should be noted, that as a result of this adaptation, not only the distributed grid components may be used, but all generic algorithms and data components developed on top of the interface are immediately available. The techniques discussed so far have been successfully used to implement distributed solvers for the 2D Euler equations for one- and two-component systems. These solvers use a fluctuation-splitting algorithm on unstructured grids with a second-order recovery step, and an explicit time integration. The mathematics are described in [SBB97]. The implementation of this solver is based on a C++ -realization of the generic grid interface mentioned above. As such, the parallelization of the code essentially involved substitution of grid functions with their distributed counterparts, and addition of synchronization calls. Also, the maximal stencil of the algorithms used has been determined by simple pencil-and-paper inspection. Due to a parameterization of the solver over the actual grid type, the same numerical code can be used both for the parallel and the sequential version.
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Currently, the parallelization of a 2D Navier-Stokes solver with implicit time integration and a two-level grid is underway. This code has been developed using its own grid data structure; thus an adaption to the generic interface had to be done. 5. Conclusion We have presented a novel approach to the development of parallel applications involving grid-based algorithms of data-parallel type, such as used in solvers for partial differential equations. In particular, the concepts allow the programmer to concentrate on the abstract aspects of grid distribution. It offers automatic generation of grid overlap based on the stencil of the client algorithms, and is therefore directly applicable to a broad class of applications. Furthermore, a generic implementation of these concepts make it easily usable. 6. Bibliography [Bas94]
Bastian, P. Parallele adaptive Mehrgitterverfahren. versitat Heidelberg, 1994.
PhD thesis, Uni-
[BB99]
Bader, G. and Berti, G. Design principles of reusable software components for the numerical solution of PDE problems. In Hackbusch, W. and Wittum, G., editors, Concepts of Numerical Software. Vieweg Verlag, 1999 (to appear).
[Ber99]
Berti, G. Generic software components for Scientific Computing. PhD thesis, BTU Cottbus, in preparation.
[Bir98]
Birken, K. Ein Modell zur effizienten Parallelisierung von Algorithmen auf komplexen, dynamischen Datenstrukturen. PhD thesis, Universitat Stuttgart, October 1998.
[Els97]
Eisner, U. Graph partitioning - a survey. Technical Report SFB393/97-27, Technische Universitat Chemnitz, December 1997.
[SBB97]
Schenk, K., Bader, G., and Berti, G. Analysis and approximation of multicomponent gas mixtures. In Feistauer, M., Kozel, K., and Rannacher, R., editors, Proceedings of the 3rd Summer Conference Numerical Modelling in Continuum Mechanics, Prague, 1997.
[Zom96]
Zomaya, A. Y., editor. Parallel and distributed computing handbook, chapter Scientific Computation. McGraw Hill, 1996.
Performing parallel direct numerical simulation of two dimensional heated jets. S.Benazzouz, V.G.Chapin &; P.Chassaing
Laboratoire de Mecanique des Fluides EN SIC A, Toulouse FRANCE
ABSTRACT Parallel direct numerical simulations of a two dimensional heated jet is performed. The relevance of a decomposition domain strategy is shown. A performances comparison between a parallel vectonal computer (FUJITSU VPP300) and a parallel superscalar one (IBM SP2) is done. Key Words: Parallelization, acceleration factor, communications, processus, vectorial architecture.
1. Introduction Our work consists in studies about the self excited global mode which develop in two-dimensional heated jet when the density ratio between the jet and the ambiant fluid is sufficiently low. We need many long and hard simulations to analyse the physics of the problem. Simulations done on single processor become too long. Our objectives by using parallel computations is to reduce the return time (CPU time) of our simulations with attention to limit the increase of the total computing time (CPU time * processors number). The second objective is to increase the size of the tractable problem in doing a memory partition. The paper will be organized as flollow. The next part shows the numerical methodology, the third part of our work explicit the optimization strategies. Part number four shows our results. The last part derives some conclusions from our results.
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Figure 1: Computing domain 2. Numerical methodology
Our code resolves on Euler equations with added fourth order numerical dissipation. The temporal scheme is a 5-stage second order explicit RungeKutta scheme [Jameson, Schmidt et Turkel (1981)] with a CFL number of 3. The spatial discretization is based on second order central differences. The boundary conditions based on a monodimensional characteristic method are following ones : reservoir conditions at the inlet, slip conditions at the lateral boundary and s non-reflecting conditions at the outlet [Hedstrom (1979)]. The initial jet field has been choosen following [Yu et Monkewitz (1990)] paper. It consists in an hyperbolic tangent profil applied on the density and the longitudinal velocity. The characteristic non-dimonsional parameters are the velocity ratio A = ^/T^00 = 1 and the density ratio which is choosen in the rangeO.25 < S — — < 0.55 for all simulations presented in this paper. Our grid has two zones: a first one named the uniform zone, with a constant space between gridpoints and a second one, named the geometric zone, with a geometric progression between each gridpoints. The region of physical interest is the uniform one. We made this in order to make our simulations cheaper in computation time and to smooth unphysical reflection from the outlet boudary to the region of interest. Figl. resumes boundary conditions and details of our grid structure.
3. Optimization strategies
The main goal of any parallelization method is to reduce the CPU time without a too high degradation of the global performance which will be analysed late on through the acceleration factor. Because the code is explicit, this made relatively easy its parallelization. Every processus needs only data located in
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Figure 2: Computing domain decomposition his subdomain except some frontiers data. Differents methods are actually used. We have choosen a multidomain decomposition strategy. This method consists in affecting a part of the global domain for each processus. This method can be also called SIMD (Single Instruction, Multiple Data) with distributed memory. Each processor computes into his own subdomain and communicate only frontiers data with neighbor processors. Different solutions exist to decompose the global domain. The decomposition can be based on longitudinal or transversal subdomains or a mixing of both methods. In order to optimize our parallel solver, we have to reduce the communication time. Our grid contains more gridpoints in longitudinal direction (TV/ > N J ) . For thoses reasons our calculation domain is decomposed through the longitudinal direction. Due to the nature of our grid, subdomains have the same number of gridpoints if the processors frequencies are the same. If the processors frequencies are not the same a simple load balancing is used through an adaption of the gridpoints number for each subdomains. Note that -in the case of same frequencies- due to our grid, each processus compute the same gridpoints but not necessarly the same field area. Fig.2 show a schematical decomposition of the global domain. To analyze the performances of our parallel code we use a parameter noted MP called the acceleration factor:
where Tm is the multiple processus running time (for each processus) , Ts the single processus running time and Np is the processus number. In a parallel computation the total time (Tm) is in general composed by communication time Tcomm and computation time Tcomp. Following this it becomes :
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Finite volumes for complex applications Meshpointe neecbd by processes P in SCHEMA
Mesh points needed by processus P inDISSIP
Figure 3: Frontiers treatment Where Tcomp can be modelized -in a domain decomposition method- by
Equation (1) become
3.1 Subdomain communication Due to the nature of our performing strategy, each processus needs to communicate with at least another processus. All processus, except the processus which compute the inlet and the outlet subdomain send and receive updated data to and from the previous and the next subdomain processus. As shown in fig.3 only subroutines SCHEMA and DISSIP need data located outside the subdomain limits to compute derivates of the physical quantities. Almost all the communication time (Tcomm) consists in those data communications. We have choosen to apply non-blocking sending communications and blocking receving communications in order to reduce the communications time which is capital in optimizing the acceleration factor.
3.2 Vectorization method The parallel vectorial version of our code is based on vectorization of the largest DO loops. The main difficulty in vectorizing a code is to make all
Complexity, performance and informatics Number of processus Time (minutes)
1 642
4 173
8 87
16 44
32 22
667
64 12
Table 1: Return times
DO loops explicit. In our solver, the main computing time is consumed into subroutine SCHEMA and DISSIP, where the DO loops are effectively explicit and hence ready for vectorization 4. Performance evaluation
All results will be shown with the same number of gridpoints in each subdomains because the processor frequencies are the same.
4.1 Acceleration factor versus processors number All experiments presented in this section are made with a grid defined by NI = 285, NJ = 213 and where uniform zone dimensions are 0 < X/D < 5 and -1 < Y/D < I In a first part we will show the dependence of the acceleration factor with the processor number. Our experiments show two important results. The acceleration factor is relatively high Mp > 0.9 and it decrease versus number of processors. Table 1 shows return time results and Fig. 4 shows the evolution of the acceleration factor with the processus number.
From equation (4) the acceleration factor can be written :
We can see clearly that the increasing of the processus number has some negative effect on the acceleration factor , this phenomena can be largely explained by the previous equation. In fact, the communication time are widely independent on the processors number so the ratio Tgym is constant. It is just depending on transversal number of gridpoints (NJ), because almost all communications are vectors of length NJ or 2*NJ.
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Finite volumes for complex applications Dependence of acceleration factor with the processors number
Figure 4: Acceleration factor 4.2 Grid size
effect
In this part all results are based on experiments made on 16 processors. A second part will the influence of the grid size on the acceleration factor. We can note that the length of each subgrid (NI/NP) has more important influence than grid width (NJ). In a first approximation we can modelize the communication time by :
And the single processus time by :
Equation (5) become
The direct dependence of the acceleration factor with NI is verified in fig.5 with a constant value for Np (Np = 16) This result is perfectly in agreement with the global domain decomposition strategy which is a longitudinal decomposition.
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Evolution of the acceleration factor
100
200
300
400
Longitudinal gridpoints number (Nl)
Figure 5: Grid size effect Computer Time (minutes)
SP2
173
VPP300 45
VPP300 with vectorial performin 16
Table 2: Return times
4.3 Vectorial performances A last part compare performances between a parallel vectoriel computer and a parallel scalar one. We notice that for the same number of processors, it is more profitable to compute on a vectoriel architecture. This comes primarily owing to the fact that the most computation time is consumed in the largest DO loops which are easily and high performed vectorisable loops. The power of vectorial architecture come from his hardware SIMD, unfortunately all loops which want to be vectorizable have to be explicit. As we see in Fig.6 the acceleration factor of the VPP300 (this computer has just 6 processors) is near the acceleration factor of the SP2. But, we obtained a return time divided by three by using vectorial instructions
5. Conclusion
We saw the performances dependence on many factors. With return time reducing, another effect of the parallelization is the memory profit, in this case large and hard simulations have to be computed on a parallel computer with attention to adapt it to the grid size.
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SP2/VPP300 Comparison
number of processors
Figure 6: Comparison SP2/VPP It is more profitable to decompose a computation domain through only one direction, in order to minimize the communication time and made more profit in return time. It is best to decompose through the direction which have the greater gridpoints number. However, for computations which are not requiring large memory, but are expensive in computation time, our interest is to used a weakly parallel vectorial architecture. This numerical study shows the best way to optimize our parallel code, how to choose a decomposition strategy and what kind of architecture to use.
Bibliography [HED
79]
[JAS 81]
HEDSTROM, G. W. Nonreflecting boundary conditions for nonlinear hyperbolic systems, J.C.P. 30, p222-237 (1979) .
JAMESON, A., SCHMIDT, W. ET TURKEL, E. Numerical solutions of the euler equations by finite volume methods using runge-kutta time-stepping schemes, A.I.A.A. paper 81-1259 (1981).
[YUM
90]
Yu, M.H. ET MONKEWITZ, P.A. The effect of nonuniform density on the absolute instability of two-dimensional inertial jets and wakes, Phys. Fluids A 2(7), July 1990, pi 175-1181 (1990).
Two-Dimensional Riemann Problems Assessment Tests for Upwind Methods for Multi-Dimensional Supersonic Flow Problems
J. van Keuk, J. Ballmann Lehr- und Forschungsgebiet fur Mechanik der RWTH Aachen Templergraben 64, 52062 Aachen, Germany
ABSTRACT Numerical simulations of two-dimensional Riemann problems using different upwind methods are presented in this paper. These problems are well suited to analyze the different methods with regard to robustness and accuracy, since they include many complex interaction phenomena as they can also appear in applied problems like super- and hypersonic intake flows. Some positive and negative features of the different methods are shown and, finally, first results of a three-dimensional intake flow simulation using one of the favorite schemes are presented. Key Words: two-dimensional Riemann problems, upwind-methods, robustness, accuracy, limiters, grid refinement, multi-dimensional supersonic flow problems.
1. Introduction The purpose of the authors's research project is to analyze three-dimensional supersonic and hypersonic flows in intakes of airbreathing propulsion systems, which are characterized by complex interaction phenomena between shocks, contact discontinuities and expansions. These all may occur simultaneously e.g. in the intake's corners. To resolve these phenomena in a physically reasonable way the use of upwind methods is necessary, but it is not sufficient to concentrate the choice of the method only on one or the other phenomenon or interaction. In a paper by P. D. Lax and X.-D. Liu [LAX 98] two-dimensional Riemann
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problems with 19 genuinely different solutions were analyzed. These problems are well suited to investigate upwind methods regarding their positive and negative features in connection with robustness and accuracy. To pick out the best methods available for the simulation of the flow problems mentioned above is the purpose of this paper. 2. Physical Model
The Navier-Stokes Equations for three-dimensional flow form the basis for the numerical simulations planned in the authors's research project. For a simply connected, three-dimensional domain V with boundary dV these equations can be written as
where U = (p, pu, pv, pw, pE) is the vector of conservative variables. Fc, Gc and Hc represent the convective flux functions and Ft,, Gv, Hv are the corresponding diffusive ones. The vector of heat conduction is modeled by Fourier's law and Sutherland's formula is adapted for the laminar viscosity as a function of the static temperature. The description of the heat conductivity is carried out via a constant Prandtl number and the gas is assumed to be thermally and calorically perfect. 3. Numerical Method
3.1 Basic Flow Solver The DLR Navier-Stokes Solver FLOWer (Version 114.1) forms the basis for the simulation of the flow problems considered in this paper. This code is formulated as a cell vertex centered finite volume scheme in a block structure and it has been extended by different upwind methods (e.g. van Leer / Hanel, AUSM, AUSM+, AUSMDV, AUSMPW, LDFSS, HLLE, Roe (Marten / Yee)) to better capture the directed propagation of information inherent in the inviscid part of the equations [VKE 98]. Some of them will be described more precisely in the next section. Central discretization is applied to the diffusive part according to its parabolic character in space and time. Time integration is performed by an explicit 5-Stage Runge Kutta scheme. For the simulation of flow problems with asymptotically steady state solutions, convergence acceleration techniques like local time stepping and the well known
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multigrid method in FMG-FAS formulation are available in the FLOWer-Code.
3.2. Upwind Methods A large variety of different upwind methods has been presented in the last 20 years. Basically, one distinguishes between flux vector splitting (FVS) and flux difference splitting (FDS) techniques. The simpler variant is to consider only the Eigenvalues of the flux' Jacobian of the inviscid flux vector and to split the flux terms according to the sign of the associated propagation speeds. This is done in FVS schemes. Sophisticated methods to introduce more physical properties into the numerical scheme are the FDS methods, where local Riemann problems at the cell interfaces are solved either exactly or approximately. The upwind methods described below are first order accurate in space. This does not suffice for the numerical solution of the Navier-Stokes Equations for reasons of consistency, because of the second order derivatives appearing in the diffusive terms; it is therefore necessary to increase the order of accuracy in space concerning the discretization of the convective terms. One possibility to formally achieve second order accuracy in space is to use MUSCL-Extrapolation (Monotonic Upstream Scheme for Conservation Laws). The idea is to consider a piecewise linear distribution of the variables ( conservative or primitive ones) within each control volume, see van Leer [VLE 79] This leads to the following expressions for left and right extrapolated values of any scalar flow variable U:
where
and $(•) are the so-called "limiter functions" to guarantee the TVD-property of the scheme. Widely used limiters were proposed by van Leer [VLE 79] and Roe [ROE 81] ("minmod", "superbee"):
The interface values can be considered as resulting from a combination of forward and backward extrapolations. The parameter K switches between these possibilities. 3.2.1. van Leer Flux-Vector Splitting Van Leer imposed continuous flux functions for the whole range of Mach numbers and eigenvalues of F^±1; which are either positive/negative or zero. i+k'
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In the subsonic regime this leads to the following expression for the split fluxes as functions of the Mach number with a second order polynomial in a slightly modified formulation by Schwane and Hanel [HAE 89].
where
a* = -y/2(« — !)/(« + l)Ht is the critical speed of sound. There is no splitting in supersonic flow since all eigenvalues have the same sign:
3.2.2. A USM Flux-Vector Splitting
(Liou/Steffen)
The Advection Upstream Splitting Method was originally proposed by Liou and Steffen in the early 90's [LIO 93]. First, the inviscid flux vector is split into an advective and a pressure term. These two terms are then split separately, leading to the following expression for the flux at the cell interface
where A i+ i{•} = {»}i+i - {•}{ and Mi+i - M+ + Mi+1. The split Mach numbers M^ are definded according to van Leer's splitting as described above. The pressure flux terms are assumed to be governed by acoustic wave speeds. An expression using second-order polynomials of the Mach number is proposed for the pressure splitting:
3.2.3. AUSMDV Flux-Vector Splitting
(Wada/Liou)
Yet another approach for the splitting of the inviscid flux vector as a mixture of a more FDS based scheme and a more FVS based one was proposed by Wada and Liou [WAD 94] and is called AUSMDV.
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where s is a switch as a function of the local pressure gradient.
It is clear to see that the AUSMD and AUSMV differ only in the treatment of the term pu2 in the x-momentum flux. The corresponding expressions are:
The velocity splitting within the AUSMDV is similar to the original proposal of van Leer extended by terms designed to capture stationary and moving contact discontinuities.
where
and cm = max(cL,CR). The pressure splitting is the same as in the original formulation and finally, the mass flux for the AUSMDV is
3.2.4. HLLE Flux-Difference
Splitting
The most simple flux difference splitting scheme was proposed by Harten, Lax, van Leer and modified by Einfeldt [EIN 91]. The solution of the whole Riemann problem is replaced by a model consisting of three constant states separated by two shocks yielding for the flux at the cell interface
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where the X^ denote the speeds of the fastest and slowest wave in this model. 3.2.5. Roe Flux-Difference
Splitting
A more sophisticated flux difference splitting scheme was developed by Roe [ROE 81]. Roe replaces the approximation of the solution of the nonlinear Riemann problem by the exact solution of the linearized problem that has to be extended by the approximation of discontinuous solutions.
[18] \k is the propagation speed of the k-th wave of the linearized Riemann problem and U,rfc are the corresponding left- and right-eigenvectors. The so constructed formulation of the flux function consists of a central part supplemented by an upwind term, which has to be computed using average values of the conservative variables. One possibility for the construction of these average values is Roe-aver aging, see Grotowsky [GRO 94]. A problem appears if one of the eigenvalues changes its sign. For centered expansion fans with sonic point the scheme then leads to an expansion shock and generates non-physical solutions such as the "carbuncle phenomenon" when calculating hypersonic blunt body flows. To circumvent this difficulty Harten proposed a modification of the modulus function in Eq.(18)
where 6 is a small number often referred as "entropy fix". For this method there exists an alternative way of increasing the formally order of accuracy in space, that was originally proposed by Harten and Yee. This Ansatz comprises an approximation of the truncation error of the first order scheme and is known as "modified flux approach" [GRO 94].
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4. Results First, two-dimensional Riemann problems as they have been investigated by P. D. Lax and X.-D. Liu [LAX 98] are considered. In these problems a quadratic solution domain is divided up into four quadrants and the initial data are constant in each of them. The initial conditions are restricted such that only one elementary wave, a shock, an expansion, or a contact discontinuity appears at each interface. Depending on the initial conditions complex interactions between the different waves are evolving in time. Fig.2 shows the results obtained with the different upwind methods for the problem number 13. The initial data for this problem consist of two stationary contact surfaces and two moving shocks. The computational domain has been discretized with 300 grid points in each direction in an equidistant manner. The density distribution is displayed. Obviously that the overall best results were obtained with the Harten / Yee FDS and the AUSMDV in connection with van Leer's flux limiter. The moving shocks are sharp, the stationary contact discontinuities correctly captured. Also the secondary contact surfaces and the small vortex structure in the center are fairly resolved. Use of the original AUSM scheme leads to spurious oscillations behind shocks and contacts due to the non-monotonous behaviour of this scheme across discontinuities. Finally, the van Leer / Hanel FVS and the HLLE are characterized by the unappropriate capturing of stationary contact surfaces and, in addition, as a consequence of the higher amount of artificial viscosity inherent to both schemes the resolution of the secondary contacts and the small vortex structure is worse, too. Fig.3 shows an investigation of the influence of three different flux limiters (Roe's "minmod" and "superbee" limiter and the one of van Leer) on the results for two different problems, what is often neglected, and a surprisingly strong influence is obvious that was not expected. In dependence of the limiter location within the TVD region a dramatic improvement of the resolution of the moving contacts in the Lax 5 problem and a better resolution of the triple points in the Lax 12 was detected. In addition, in both cases the use of the "superbee" limiter leads to small oscillations as a consequence of the vicinity to instability of this limiter, so that perhaps the van Leer limiter should be the best choice. As some kind of a grid sensitivity analysis Fig.4 shows solutions for the Lax problem number 5 on a refined grid (700 grid points in each direction) and in nice agreement with the work of Lax and Liu small vortex structures appear that could not be resolved with any method on the coarser grid. The fact, that the Harten / Yee FDS resolves only one vortex structure may be due to some kind of a direction dependency in the way of increasing the formal accuracy in space. In conclusion again the influence of the flux limiter is clear to see when looking at the pressure distributions obtained with the van Leer / Hanel FVS. Use of van Leer's limiter yields a vortex resolution almost comparable to the AUSMDV solution whereas the "minmod" limiter nearly prevents any vortex to appear.
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Finally, Fig.l shows first results of a threedimensional intake flow simulation using the Harten / Yee method. On the upper left hand side of this figure the result for the density distribution for the two-dimensional case is plotted. For the three-dimensional case the interior part of the intake is assumed to have sidewalls beginning at the intake lip. On the lower left hand side of the figure the density distribution in the symmetry plane of the inlet is showed. The phenomena in front of the intake lip are obviously exactly the same as for the two-dimensional case, but also the shock positions and angles as well as the separation region on the upper side of the interior part are at least similar, because the three-dimensionality does only less influence the effects in the symmetry plane. Finally, on the right hand side of the figure two cross sections in the interior part are plotted, which show the three-dimensional effects in the lower corners produced by the two shocks originating from the intake lip and the sidewalls. 5. Conclusions
Numerical solutions of 19 two-dimensional Riemann problems using different upwind methods have been realized. Three of them are presented. The overall best results were obtained using the Roe FDS in the Harten / Yee formulation and the AUSMDV. The original AUSM scheme showed a nonmonotonous behaviour across discontinuities resulting in pressure and density oscillations behind them. Furthermore, a surprisingly high influence of the flux limiter was detected. So deficiencies of the schemes exhibiting more numerical viscosity could partly be cured by an appropriate limiter. Finally, first promising results of a three-dimensional intake flow simulation using the Harten / Yee method were presented. Unfortunately a considerable high value for the entropy-fix had to be chosen for reasons of stability, so that subject of future work in this project will be simulations using the AUSMDV, since it proved to be comparable accurate in the resolution of even complex interaction phenomena at a lower computational cost and higher stability. 6. Bibliography
[LIO 93]
Liou M.-S., STEFFEN C. J., «A New Flux Splitting Scheme », J. Comp. Phys. 107, pp. 23-39, 1993.
[EIN 91]
EINFELDT B., MUNZ C. D., ROE P. L., SJOGREEN B., «On Godunov-Type Methods near Low Densities », J. Comp. Phys. 92, pp. 273-295, 1991.
[GRO 94]
GROTOWSKY I. M. G., «Ein numerischer Algorithmus zur Ldsung der Navier-Stokes-Gleichungen bei Uber- und Hyperschallmachzahlen, PhD Thesis, RWTH Aachen, 1994.
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[HAE 89]
SCHWANE R., HANEL, D., «An Implicit Flux-Vector Splitting Scheme for the Computation of Viscous Hypersonic Flow », AIAA Paper 89-0274, 1989.
[LAX 98]
LAX P. D., Liu X.-D., «Solution of Two-Dimensional Riemann Problems of Gas Dynamics by Positive Schemes », SIAM J. Sci. Comput. 19, No. 2, pp. 319-340, 1998.
[ROE 81]
ROE P. L., «Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes », J. Comp. Phys. 43, pp. 357372, 1981.
[VKE 98]
VAN KEUK J., BALLMANN J., SCHNEIDER A., KOSCHEL W., «Numerical Simulation of Hypersonic Inlet Flows », AIAA Paper 98-1526, 1998.
[VLE 79]
VAN LEER B., «Towards the Ultimate Conservative Difference Scheme V. A Second Order Sequel to Godunov's Method », J. Comp. Phys. 32, pp. 101-136, 1979.
[WAD 94]
WADA Y., Liou M.-S., «A Flux Splitting Scheme with High-Resolution and Robustness for Discontinuities », AIAA Paper 94-0083, 1994.
Figure 1: 2D/3D Intake Flow Simulations
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Figure 2: 2D Riemann Problem according to Lax Problem 13 (Two Shocks/Two Contact Surfaces)
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Figure 3: Influence of the Flux Limiter for Lax Problems 12 (Two Shocks/Two Contact Surfaces) and 5 (Four Contact Surfaces)
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Figure 4: Effects of Grid Refinement and Flux Limiters
Robustness and accuracy on unstructured grids Numerical experiments on finite volume schemes
Ernst A. Meese
Skjalg E. Haaland
CFD norway as Teknostallen Professor Brochsgate 6 N-7030 Trondheim, Norway
Department of Applied Mechanics, Thermo and Fluid Dynamics NTNU N-7491 Trondheim, Norway
ABSTRACT With the two-dimensional incompressible Navier-Stokes equations in mind, the focus of the present work was on finite volume schemes on triangular and quadrilateral unstructured grids. Numerical comparative studies were performed on selected schemes to determine the schemes' robustness of accuracy with respect to grid quality. The tests were performed on large series of randomly distorted grids with varying degree of distortion. Cell centred and vertex centred discrete diffusion operators were considered. A vertex centred scheme had the lowest complexity of construction for second order accuracy. The convection operators were currently limited to vertex centred schemes. Various interpolations and locations of quadrature points were studied. Some superconvergent behaviour on strongly distorted grids was observed for both diffusion and convection operators. Lastly, an unstructured non-linear flux limiter was developed from established structured methods. It was tested and found successful. Key Words: finite volume, unstructured grid, experiments on accuracy, superconvergence, cell centred, vertex centred, quadrilateral cell, triangular cell, flux limiter
1. Introduction It is common to judge the accuracy of a finite volume scheme by an analysis of the order of it's truncation error. This is, however, not sufficient on general grids. The truncation error may be very poor due to grid skewness, but good solutions may still be obtained. Hard analysis of the global error is usually out of reach. Hence, numerical comparative studies are necessary to determine the relative performance of the schemes, particularly because such studies seem to
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be rather rare in the literature. It is common to assess the quality of finite volume schemes by numerical experiments on a set of well established benchmark problems. These benchmark comparisons are often formulated on quite smooth grids. However, unstructured grids are often generated automatically, and must be expected to contain quite distorted cells in some regions. Distorted grid cells may greatly reduce accuracy, and it is our opinion that tests on smooth grids are insufficient for the general verification of a finite volume scheme. Therefore, the present comparative studies were performed on randomly distorted grids. Methods performing well on such extreme grids may be used with confidence under more regular conditions. Two strategies for locating the unknowns were considered in the present work. In the cell centred (CC) (primary grid) strategy the unknowns are located at the centre of each grid cell. In the vertex centred (VC) (secondary or dual grid) strategy, the unknowns are located at the vertices of the grid cells, and new control volumes for the equations are formed around each vertex. Which of the two strategies to be considered best for general purposes are under dispute in the literature [Mav 97]. The schemes considered here all have truncation error worse than second order on distorted grids, but they have truncation error to second order on uniform grids. Arguments have been made that such schemes really are second order [TMa 85, TDB 96]. However, these arguments assume the existence of a grid distribution function x — x(£) = g(£/N) that is independent on N, in which case the local stretching parameter r = hi+i/hi will go uniformly to 1 as N goes to infinity, and the truncation error will eventually show second order behaviour. In real application, r and skewness are seldom very small, and low order behaviour should be expected. However, it is not truncation error, but global error that we want to reduce. Standard theory states that the global error is never worse than the truncation error, but there is room for it to be better, and this is in fact the case for several simple schemes, however, not for all. In the following, some finite volume schemes relevant for the convection and diffusion operators of the incompressible Navier-Stokes equations and the test results on them are presented. A non-linear flux limiter is also tested. Due to the space limitations the presentation is quite brief, but a thorough presentation may be found in [EAM 98]. 2. Discretisation of the diffusion operator
The diffusion operator of the incompressible Navier-Stokes equations may be stated as
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where the sum is over the edges of a polygonal shaped control volume Q and HJ is the outward unit normal vector at edge £j. The line integrals along each edge may be approximated by any preferred one-dimensional quadrature. For the CC schemes we use the midpoint rule along each edge. Assuming a constant v, we need an estimate of V<£> to complete a finite volume scheme for the operator (1). The Naive Cell Centred (NCC) method is simply to use the difference between the values of
The control volume Q is selected as the polygon formed by the six nodes at the centres of the cells in the immediate vicinity of the edge e. In compass notation on regular quadrilateral grids this amounts to the nodes P, E, N, NE, S, and SE. The extension to triangular cells is trivial. To form the rightmost integral of Eq. ( 2 ) , the trapezoidal rule is used along each line segment of f2. A drawback of this Cell Centred scheme with Integral evaluation of the gradient (CCI) is that it becomes quite complicated for the grid edges sharing a vertex with a boundary edge. In this situations one or more of the six neighbouring nodes are missing and some other fi has to be formed. The last cell centred scheme we considered was based on a least squares approximation of the gradient (CCLS). Our formulation is motivated by a need for the formation of the matrix in an implicit method. Other methods are suggested for explicit methods [Bar 89]. The estimate is formed by the linear combination
where the sum is over the nine neighbouring nodes P, W, E, SW, S, SE, NW, N, and NE using compass notation. Only regular quadrilateral grids were considered, and these nine nodes are used for all four edges of the cell under consideration to get a compact stencil. This makes the gradient estimate different for a given edge when viewed from its two adjacent cells, but this has posed no problem. Using Taylor series analysis we may find six equations for the nine coefficients of Eq. (3) to satisfy for second order accuracy of the gradient estimate. To make the solution unique, we aim to minimise the expression
using the six equations mentioned above as side constraints.
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Figure 1. Details of the subdivision of primary grid cells into secondary cells. The shaded area is the part of the secondary volume around vertex 1 contained in C. Finally, the vertex centred scheme (VCI) based on Eq. (2) was considered. A control volume is formed around each vertex by drawing lines from the centre of each edge to the cell centre (see Fig. 1). A gradient V
3. Discretisation of the convection operator
The part of the integral of the convection operator around the vertex 1 in Fig. 1 residing within the primary cell C is
where wf,
where for a primary cell d with m, vertices. Four locations for the quadrature points were considered. Referring to Fig. 1, on uniform grid these points amounts to x the midpoint scheme over each secondary edge, A the midpoint scheme on each secondary edge segment within the primary cell, -f in the fourth order Gaufiian quadrature over the secondary edge, and Q in the trapezoidal rule over each secondary grid edge. Two means of selecting the coefficients Cjk of Eq. (6) was used. Firstly, the coefficients obtained on a uniform primary grid can be used also on deformed cells. We call this the constant coefficient schemes. Secondly, it is also possible
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to obtain linear reconstruction within each primary cell. Small equation systems (3 x 3 for triangular and 4x4 for quadrilateral cells) are obtained for the coefficients Cjk by Taylor analysis, demanding a second order truncation error of the interpolation. These systems are solved numerically for each primary cell. The flux limiter was implemented on the scheme with u^ located at A and 0j* at x. However, instead of finding <^f by the arithmetic average, an interpolation 0f = ai + (1 — a}4>2, & € [0, 1], was used assuming the flow along the edge e\ to be from vertex 1 to vertex 2. For a > 1/2 the scheme becomes upstream weighed. On structured grids, it has become commonplace to adopt the normalised variable 0c — (
was developed for this purpose [EAM 98]. 4. Numerical Experiments
The random grids were generated by disturbing the uniform grid mesh length h into hi = 1 — c/rnd#, for edge i, where rnd#; £ [0, 1] is some random number. When d = 0 there is no disturbance and for d — 1, any grid length might be the result. The uniform mesh length is here normalised to 1. A uniform triangular grid consists of only equilateral triangles. Some smoothing is necessary on the most distorted grids to ensure valid cells. In all results presented here, the maximum norm of the global error versus the largest edge length in the grid is used. 4.2 Experiments on the diffusion
operators
To study the isolated properties of the discretisations for the diffusive terms of the incompressible Navier-Stokes equations, Poisson equations are feasible,
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Figure 2. NCC on Eq. (8) on quad. Figure 3. NCC on Eq. (8) on triancells gular cells
Figure 4. CCI on Eq. (8) on quad, Figure 5. CCLS on Eq. (8) on quad, cells cells
Figure 6. VCI on Eq. (8) on quad, Figure 7. VCI on Eq. (8) on triangular cells cells
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A test case looking somewhat like a top hat, was proposed. The solution is
and the right hand side function / of (8) is selected to fit this solution. The domain of definition is ( x , y ) £ [0,1] x [0,1]. The "Top Hat" Poisson equation is designed to study discretisations at the interior domain, as the flat solution at the boundaries gives little influence on the global error from boundary discretisation errors. Furthermore, the rather sharp gradient simulates situations met in fluid flow problems. All methods tested showed good second order accuracy on rectangular grids. However, the NCC scheme is seen to perform very poorly on skewed grids (Figs. 2 and 3). The CCI scheme is somewhat better (Fig. 4), but we have to go to the CCLS scheme go get second order accuracy (Fig. 5) for a cell centred scheme. More surprising is the second order of the VCI method (Figs. 6 and 7). The low order truncation error is hence of a kind that do not manifest itself into the global error. The VCI scheme is considerably less complex than CCLS, but yields comparable results. 4.3 Experiments on the convection operators The velocity field for the convection was the solid body rotation u(x, y) = y and v ( x , y) = —x over the domain (x, y) £ [—1,1] x [0,1] with inlet and outlet on the boundaries y = 0 and — 1 < x < 0 and 0 < x < I respectively. The equation solved was (r = y .r2 + y2)
with solution
Some results are presented in Fig. 8. Runs were also made on rectangular grids, in which case all schemes were second order accurate even for very large aspect ratios. The best of the constant coefficient schemes for convection are not superconvergent to second order, however, some superconvergence is in effect. Really, all the schemes has zero'th order truncation terms, but the average global error lies between first and second order for most cases. Looking at the results for the variable coefficient schemes, it seems that some of this superconvergence was lost on quadrilaterals. The only scheme that is linearity preserving is the one with both u and 0 represented at A. This scheme has the best order estimate of all tested schemes, however, the average error level is 2-3 times higher than the best constant coefficient schemes.
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Figure 8. The plots show curve fits to the scatter maximum global error data for each combination of reconstruction of u and . The vertical spread of these scatter data is quite large, indicating significant dependence on grid skewness. The two top plots are for constant coefficient schemes, the two bottom for linear reconstruction schemes, the two left most on quadrilateral grids, and the two rightmost on triangular grids.
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Figure 9. VCI for Eq. (12) on quads Figure 10. VCI for Eq. (12) on triwithout flux limiting angles without flux limiting
Figure 11. VCI for Eq. (12) on Figure 12. VCI for Eq. (12) on quads with ACUTER quads with ACUTER
4.4 Experiments on a convection-diffusion
equation
To study the effect of combining the convection and diffusion operators, and also to study the flux limiting, the equation
over the unit square was selected. The results presented in Figs. 9 through 12 are for v — 1/20. In this case, the solution has a "boundary layer" along the lines x — 0 and y =. 1 and flux limiting is essential on coarse grids, and we actually see that ACUTER, in addition to giving monotone solutions, yields somewhat better results than central differences even on fine grid. The distortion level is labelled as; 0 - Uniform, o - 5%, D - 10%, o - 15%, x - 20%, + - 35%, A - 55%, V - 90%. The dotted lines indicate the second order slope.
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The data show a clear second order accuracy trend, and the vertical spread is small, indicating small dependence of the accuracy on the grid cell skewness. This is opposed to the pure convection case.
5. Conclusions
Experiments on large sets of random grids have revealed properties of the global error of the schemes under study. Superconvergence to second order was demonstrated for the CCLS and VCI schemes on the diffusion operator, and also for the VC diffusion-convection scheme both with and without flux limiting. The flux limiter ACUTER provided both monotone solutions on coarse grids and some enhanced accuracy. The VC convection operators are less robust with respect to grid skewness, but it is interesting to note that constant coefficients proved to be just as good, or even better, as linear reconstruction. It seems clear that neither primary nor secondary grid methods, as such, could be claimed to be the more accurate. A more relevant question, although still not precise, is with what simplicity or ease a prescribed accuracy may be achieved on either primary or secondary grids. In this respect the VCI scheme comes out a winner in the present tests. [Bar 89]
BARTH, T. J., Recent developments in high order fc-exact reconstruction on unstructured meshes, AIAA paper 93-0668, 1989.
[GLa 88]
GASKELL, P. H. AND LAU, A. K. C., Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm, Int. J, Numer. Meth. Fluids, 8 (1988), pp. 617-641.
[GMe 96]
GRASSO, F. AND MEOLA, C., Euler and Navier-Stokes equation for compressible flows: finite-volume methods, Handbook of computational fluid mechanics, Peyret, R., ed., Academic Press, 1996, pp. 160-277.
[Leo 88]
LEONARD, B. P., Simple high-accuracy resolution program for convective modelling of discontinuities, Int. J. Numer. Meth. Fluids, 8 (1988), pp. 1291-1318.
[Mav 97]
MAVRIPLIS, D. J., Unstructured grid techniques, Annual Review of fluid Mechanics, Annual Reviews inc., 1997, pp. 473-514.
[EAM 98]
MEESE, E. A., Finite volume methods for the incompressible Navier-Stokes Equations on unstructured grids, dr.ing. thesis, Norwegian University of Science and Technology, 1998.
[TMa 85]
THOMPSON, J. F. AND MASTIN, C. W., Order of difference expressions in curvilinear coordinate systems, J. Fluid Eng., 107 (1985), pp. 241-250.
[TDB 96]
TREGUIER, A. M., DUKOWICZ, J. K., AND BRYAN, K., Properties of non-uniform grids used in ocean general circulation models, Journal of Geophysical Research, 101 (1996), pp. 631-644.
A validation of an efficient numerical method for 3-D complex flows
E.A. Fadlun1, S. Leonardi1^, R. Verzicco2 & P. Orlandi] 1 Universita di Roma, "La Sapienza," Dipartimento di Meccanica e Aeronautica via Eudossiana 18, 00184 Roma, Italy. 2 Politecnico di Bari, Istituto di Macchine ed Energetica, Via Re David 200, 70125, Bari, Italy.
ABSTRACT A second-order accurate, highly efficient method is developed for simulating unsteady three-dimensional incompressible flows in complex geometries. This is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given surface not coinciding with the computational grid. The governing equations, therefore, can be discretized and solved on a regular mesh thus retaining the advantages and the efficiency of the standard solution procedures. Key words: 3D complex flows, Immersed boundaries,
Finite-differences
1. Introduction The continuously growing power of computers is strongly encouraging the engineers to rely on computational fluid dynamics for the design and testing of new technological solutions. Numerical simulations allow the analysis of the phenomena without resorting to expensive prototypes and difficult experimental measurements. Flows in simple geometries use regular grids and hence very efficient numerical methods. Flows in complex geometry requiring body-fitted curvilinear or unstructured meshes are less efficient and more difficult to code. In addition, in most of industrial applications, geometrical complexity is combined with moving boundaries which increase the computational difficulties since they require, regeneration or deformation of the grid.
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In view of these difficulties it is clear that a numerical procedure that can cope with the flow complexity but at the same time retains the accuracy and high efficiency of the simulations performed on fixed regular grids would represent a significant advance in the study of industrial flows. One possibility for the solution of this problem is the introduction of a body-force field f such that a desired velocity distribution V can be assigned over a boundary S. In principle there are no restrictions for the velocity distribution V and for the shape and motion of S; therefore a wide variety of boundary conditions can be imposed. The main advantage of this approach is that f can be prescribed on a regular mesh so that the accuracy and efficiency of the solution procedure on simple grids are maintained. Indeed this idea is not new since it has already been pursued in the past by methods slightly different each other but in principle almost equivalent (Peskin (1), Goldstein, Handler & Sirovich (2)). Recently Mohd-Yusof (3) has derived an alternative formulation of the forcing that does not affect the stability of the discrete-time equations. In this case the formulation of f is flow independent. The expression for the forcing f derived by Mohd-Yusof (3) is:
where RHS /+1 / 2 contains convective, viscous terms and the pressure gradient. If now we ask which value for f'"1"1/2 will yield u /+1 = V /+1 on the immersed boundary this is simply given from the above equation
An adapted version of the forcing of Mohd-Yusof (3) has been developed to combine body-forces with finite-differences.
2. Results
2.1 Vortex ring formation The generation of a vortex ring is generally obtained by pushing a finite amount of fluid through a curvilinear nozzle. The ring was generated in water and injected fluid was colored by di-sodium fluorescein illuminated by a laser sheet in a plane crossing the axisymmetric nozzle through a diameter. This provided flow visualizations that could be compared with the numerical simulations. As the fluid flows along the inner walls of the nozzle some vorticity is generated and this separates at the corner in the exhaust region. This vortex sheet is very unstable and starts rolling up forming a compact toroidal structure. During the roll up, the primary vorticity induces at the external wall of the nozzle a secondary vorticity that
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Figure 1: Comparison between numerical and experimental vortex trajectories at Re = 500 (a), Re = 1500 (b), Re = 2500 (c). filled A numerical • experimental primary ring, filled V numerical filled o experimental secondary ring. • numerical x experimental tertiary ring. also separates at the corner and forms a secondary vortex ring. This structure, has a circulation oppositely signed with respect to the primary ring, thus propagating backward inside the nozzle. Depending on the Reynolds number, i.e. on the strength of the primary vortex ring, the secondary vortex can be strong enough to generate a tertiary vortex ring that propagates in the positive axial direction. This complex behaviors is described by the trajectories of the vortex patches in 1. The Reynolds number flow dependence observed in the laboratory has been confirmed by the numerical simulations and figure 1 shows the good agreement between numerical and experimental trajectories. Before concluding this section we wish to stress an important points: the adequate description of the different flow regimes implies that the strength of the vortex structures is correctly predicted. This means that not only the velocities at the walls but also the vorticity generation is well described by the present approach. 2.2 Flow around a sphere In this section we consider the flow around a sphere where the separation is determined solely by the viscous processes at the wall. This should evidence eventual inaccuracies in the treatment of the boundary. In this flow a uniform velocity U is imposed at a certain distance from a sphere of diameter D. The Reynolds number is Re = UD/v and, depending on its value, the flow exhibits different regimes ranging from steady to unsteady up to fully turbulent. As a first test we have reproduced the Fornberg's (4) simulations; he used boundary fitted coordinates to simulate the axisymmetric flows ranging from Re = 100 up to
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Figure 2: (a): length of the separation bubble vs Re; • present results, • results by (4). linear fit. (b): pressure profiles along the sphere surface at Re = 100, 200, 500 and 1000 (for increasing pressure values); present results, data from (4) Re 100 200 500 1000
CD (Fornberg (4)) 1.0852 0.7683 0.4818 0.3187
CD (Present results) 1.0794 0.7567 0.4758 0.3209
% error 0.53 1.50 1.24 0.69
Table 1: Comparison of the drag coefficient for the flow around the sphere at different Reynolds numbers.
Re = 5000. We are aware that for Re > 300 the hypothesis of axial symmetry is not physical because the flow develops three dimensionalities in the wake, however, the high-Re axisymmetric simulations were done to validate the numerical procedure. The flow around a sphere does not separate up to Re ~ 24 and for increasing Reynolds the axial length of the separation bubble grows linearly up to Re ~ 100. This result has been found by the present numerical simulations as it is shown in figure 2. For Re > 100 the pressure distribution over the surface of the sphere and the force coefficients were calculated by Fornberg (4); figure 2b shows that also for these quantities the present numerical methods is very accurate. From the pressure and velocity field around the sphere we have computed also the drag coefficient Q that, as shown in Table 1, agrees very well with the results of Fornberg (4) and the error never exceeds the 2%. In order to test the capability of the code to handle fully three-dimensional flows and to capture flow transitions we have performed simulations in which the hypothesis of axial symmetry was removed. Similar simulations were performed by Mittal (5) using boundary fitted grids and he found that the flow preserves the symmetry for Re < 270. For 270 > Re < 300 the wake becomes wavy but it still preserves the symmetry about a plane crossing the sphere in the centre. Finally for Re > 450 the flow becomes fully three-dimensional and any degree of symmetry is lost. This is confirmed by the vorticity contour plots of figure 3 showing the oscillation of the wake only in one plane. Mittal (5) obtained essentially the same results removing any symmetry by giving a random perturbation. The last simulation, at Re — 500, which
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Figure 3: Contour plots of azimuthal vorticity at Re = 300 ((a)-(c)) and Re = 500 ((d)-(f)) through orthogonal sections. (Acoe = ±2). Grid 97 x 97 x 193 gridpoints in the azimuthal, radial and axial directions. is above the second critical Reynolds number, by the flow visualizations of figure 3 shows the loss of the previous symmetry. In fact in this case the waviness of the wake is in two orthogonal planes.
2.3 Flow stability in a channel with cavities The previous cases were simulated by plugging the body force method in a code in cylindrical coordinates. To study the flow past a cavity in a channel with periodicity in the streamwise direction the body force method has been inserted in a code that was previously used to perform the DNS of the turbulent channel flow. As a first simple case the critical Reynolds number for the plane channel has been evaluated; the value of 5850 agrees satisfactory well with the theoretical value of 5772. Ghaddar et al. (9) by a spectral element method investigated the effects of a cavity to reduce the transitional Reynolds number. They evaluated the critical Re by linear theory and by DNS; in the two cases they respectively found 900 and 1000, which agree well with the present value of 1050. To excite the unstable modes, a random disturbance at each point has been added to the steady state solution; after a transient, the velocity oscillates as V = Voea^~'(^sin(w(t ?o))- From the time history of one of these signalsCTand co can be calculated. Table 2 shows that there is a good agreement between the present values and those given in Ref. (9). By our simulations it has been investigated the variation of transitional Reynolds number as the ratio height (h)l width (/) of the cavity was varied. For h/l — 1/4,1 /3 and 1/2, respectively, the critical Re is 2758, 2138 and 1120 showing that the deeper is the cavity the stronger is the effect induced on the flow. The reason is that a deep cavity produces a thin vorticity layer in the region between the cavity and the channel. This layer rolls up and modifies the Tollmein Schlichting waves that are the typical vortical structures of an unstable flow in a channel. Figure 4 shows that in the case of a very shallow cavity the waves are very similar to those in a channel while these have
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Ghaddar et al. (9) theory linear -.043 -.0075
Present -.0419885851 -.0145530291 -.00758921262
Table 2: Comparison between the exponential decay of a of the perturbation velocity
Figure 4: Contour lines of the fluctuating component of the stream function 7 = 5 the flow moves from left to right a completely different wave number when the cavity is deeper. In this case quantitative comparisons were not possible; the study was performed because the aim is to carried out the DNS of the flow in similar geometries to investigate how the shape and the spacing between the cavities affect the near wall vortical structures. The simulations are currently running and the results will be presented elsewhere.
2.4 Flow inside a 1C piston In this last example we show the capability of the method to simulate high-^?e turbulent flows in complex geometries for which a turbulence model is needed. In addition this example includes a moving boundary and this feature was missing in the previous flows. The configuration chosen is a simplified axisymmetric piston-cylinder assembly with a fixed central valve. For this configuration experimental measurements are available for the validation of the numerical results (Morse et al. (6)). The profiles are available at 10mm increments starting from the cylinder head for crank angles 36" and 144° after top dead center. In the experiment the piston was externally motored and the valve was fixed and a tiny annular gap was left open between the valve and the cylinder head, no compression phase was included in the flow dynamics. The piston was driven by a simple harmonic motion at a speed of 200rpm ~ 21rad/s which for the present geometry yields a mean piston speed of Vp = 0.4m/s (when averaged over half cycle). The Reynolds number of the flow based on Vp and on the piston radius is Re = 2000 in air. Details of the subgrid-scale model used in this computation are given in Verzicco et al. (7). In Verzicco et al. (7) extensive analysis of the flow inside the piston has been performed showing the changes in the dynamics when the axial symmetry was enforced,
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Figure 5: Flow inside the 1C piston: Vector plots at t = 7U/2 of the oscillation cycle (a) and (b). The r— 0 section (b) is taken 10mm below the ceiling of the cylinder. Grid 65 x 65 x 151.
Figure 6: Comparison between numerical and experimental averaged vertical velocity profiles for the crank angle 36" respectively 10 (a), 20 (b) and 30mm (c) below the cylinder ceiling. present results, o experiments by (6). when three-dimensionalities developed and when the turbulence set in. In this paper we only show the turbulent case for which the comparison with the experiments was made. In figure 5 snapshots during one instant of the oscillating cycle are given and the the high three-dimensionality of the flow can be appreciated form the vector plots in orthogonal sections. Radial profiles of axial velocity were obtained by phase averaging the fields over four piston cycles and then averaging in the azimuthal direction. These profiles are shown in figure 6 for several distances from the ceiling of the cylinder at two different crank angles and the comparison with the experimental data shows that the agreement is always very good. In Verzicco et al. (7) profiles for additional sections and rms profiles of axial velocity are also given showing that the quality of the agreement is always very good. Finally in Haworth (8) the present results are compared also with the data obtained by a code using unstructured boundary fitted deformable meshes. In that paper it is shown that the quality of the results is the same even if the immersed boundary technique is incomparably less expensive.
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3. Conclusions In this paper we have summarized some of the results obtained by a numerical method based on Cartesian or cylindrical coordinates in combination with boundary body-forces. We would like to point out that the coding is very simple and in our group has been found that undergraduate students can learn to use the code in few weeks. In addition we found that it is very important to have an efficient and friendly software of interface between a CAD package and the flow solver and we are working on it. We are really confident that in a short while our group is going to produce a package to solve laminar flows past any kind of two and three-dimensional bodies. On the other hand to have a reliable package to solve realistic flows at high Reynolds numbers, it is important to spend a lot of time to test different kind of subgrid models. This is a difficult task that can not be achieved by a small group as that at the University of Roma. More collaboration is necessary and also on this point we are working together with some scholar at the Center of Turbulence Research at Stanford. [1] Peskin, C.S., 1972, Flow patterns around heart valves: a numerical method. J. of Comp. Phys., 10, 252-271. [2] Goldstein, D., Handler, R. & Sirovich, L., 1993, Modeling a no-slip flow boundary with an external force field. J. of Comp. Phys., 105, 354-366. [3] Mohd-Yusof, J. 1997, Combined Immersed boundaries/B-splines methods for simulations of flows in complex geometries. CTR Annual Research Briefs 1997, NASA Ames/Stanford Univ., 317-327. [4] Fornberg, B., 1988, Steady viscous flow past a sphere a high Reynolds number. J. Fluid Mech., 190, 471-489. [5] Mittal, R., 1999, Planar symmetry in the unsteady wake of a sphere. AIAA J., 37, 388-390. [6] Morse, A.P., Whitelaw, J.H. & Yanneskis, M., 1978, Turbulent flow measurement by Laser Doppler Anemometry in a motored reciprocating engine. Report FS/78/24, Imperial College Dept. Mech. Eng. [7] Verzicco, R., Mohd-Yusof, J., Orlandi, P. & Haworth, D.C., 1999, LES in complex geometries using boundary body forces. Proc. of the 1998 CTR Summer Program. VII, 171-186. [8] Haworth, D.C., 1998, Large-Eddy-Simulation of in-cylinder flows, in Multidimensional Simulation of Engine Internal Flows, IFP, RueilMalmaison, France. [9] N.K. Ghaddar, K.Z. Korczak, B.B. Mikic & AT. Patera, 1986 Haworth, D.C. & Jansen, K., 1997, Numerical investigation of incompressible flow in grooved channels. JFM, 163, 99-127.
Comparison of Two Finite Volume Methods for 3D Transonic Flows through Axial Cascades J.Foft, J.Fiirst, J.Halama, K.Kozel Department of Technical Mathematics CTU Prague Karlovo ndmesti 13 CZ-121 35 Praha 2
ABSTRACT This paper deals with the numerical solution of 3D inviscid transonic flow through axial turbine cascades, mathematically modeled by the system of Euler equations. Numerical solution is computed on structured grids using two different finite volume methods: simplified cell-centered TVD MacCormack scheme and cell-vertex Ni's based scheme. Two 3D axial cascades, one stator and one rotor, from last LP turbine stage of Skoda Pilsen-Turbines factory have been solved. A comparison of achieved numerical results for the stator and the rotor cascades is presented. Influence of body forces in the case of rotor flow is discussed. Key Words: Euler equations, transonic flow, turbomachinery
1. Introduction Numerical simulation of transonic flows in turbomachinery branch is subject of our long time cooperation with industry. Several numerical methods for the computation of 2D flow through cascades have been developed. Results have been thoroughly verified by measurements taken in real turbine and compressor cascades. 3D results of two independent numerical methods have been compared each other, because there are no proper experimental data available in 3D case. This enables validation of numerical results and improvement of numerical methods.
2. 2D problems and methods Inviscid flow through a cascade has been modeled by Euler equations in
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Cartesian or curvilinear coordinates. Modified system of equations describing streamtube thickness variations has also been involved. Laminar viscous flow has been modeled by Navier-Stokes equations. Following numerical methods have been employed: (a) finite volume MacCormack scheme (TVD and the classical form) (b) Ron-Ho-Ni finite volume scheme (c) multistage Runge-Kutta finite volume scheme (d) ENO and WENO schemes (e) implicit Hwang-Liu TVD scheme (f) explicit and implicit forms of Osher scheme All schemes are of cell-centered type and only (b) is of cell-vertex type. Schemes (a), (b), (c) and (d) have been used on H-type quadrilateral grid, the scheme (e) on unstructured triangular one and the scheme (f) on unstructured hybrid grid. Schemes (a), (b) and (c) are central schemes and schemes (d), (e) and (f) are upwind schemes. Schemes (a), (b) and (c) have been extended to 3D. Mentioned numerical methods have been tested and validated with experimental data obtained for different geometrical configurations. For example, typical turbine cascade SE 1050 of Skoda Turbines, Pilsen, measured in transonic wind tunnel of IT CAS, see [FOJ 94] and [FOR 94] and VKI-LS82 turbine cascade, see [HUL 92]. 2D inviscid methods combined with computation of a boundary layer have been used in turbomachinery design. Evaluated choke characteristics and loss predictions have been successfully compared with experimental data gathered by ARTI Prague. A comparison of results of unsteady flow through SE 1050 cascade, with time dependent outlet pressure, achieved by methods (a) and (b) was published in [FOJ 94]. Two techniques for the solution of modified system of Euler equations have been developed. The first one simulates the influence of AVDR (Axial Velocity Density Ratio) by modified continuity equation [FOJ 94] and the second one by using Si surface [FOJ 94] (quasi 3D approach of Wu). The second technique has been applied for the computation of flow through radial cascades. 3. 3D flow through axial blade rows
3.1. Governing equations 3D inviscid transonic flow in relative frame of reference is described by the
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system of Euler equations in non-dimensional form:
where p denotes density, Wi are the Cartesian components of relative velocity vector, e is total energy per unit volume, p is static pressure, 17 is angular velocity, K is ratio of specific heats and R is the radius. The system of equations is closed by the equation of state for ideal gas:
The right hand side of (1) is equal to zero for the flow in absolute frame of reference, i.e. with fJ = 0. 3.2. Boundary conditions The non-permeability conditions w • n = 0 (n denotes normal vector) are prescribed along walls. Stagnation values of density po and speed of sound QQ and spatial distribution of velocity angles Cj are imposed on the inlet section. Distribution of outlet static pressure p — p(R] is given. 3.3. Numerical solution Numerical solution has been computed on structured hexahedral grid by the methods (a) and (b). Equations (3) represents ID version of method (a) in absolute frame of reference. The third equation is a TVD correction.
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The modification, where v is computed using the smallest absolute value of eigenvalue instead of spectral radius PA , has lower amount of artificial viscosity (see [ANG 96]). The second method (b) approximates relation (4)
where the first two terms are first and second order convective terms and last two terms are first and second order terms including body forces. D is control volume consisting of eight cells and D is the cell of dual mesh. Artificial viscosity is implemented in the form of additional redistribution terms in each grid cell. 3.4. Numerical results Two axial cascades of the last LP steam turbine stage of Skoda Turbines, Pilsen are presented within this subsection. The first one is the stator cascade with strong divergence of the tip part. The inlet flow is of axial direction and outlet static pressure depends on radius. The results are plotted in the form of Mach number isolines on the blade surfaces (Fig.l and Fig.2). The second one is the rotor cascade with highly twisted blades, typical for the LP steam rotor cascades. Following design procedure of a rotor blade which places usually several profiles (verified in linear cascade configuration) at
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Fig.l.a: pressure side, method (a)
Fig.l.b: pressure side, method (b)
Fig.2.a: suction side, method (a)
Fig2.b: suction side, method (b)
a different radial positions, it has been proposed to consider the rotor cascade as a non-moving cascade in relative velocity field. Rotor inlet flow field parameters have been computed from the solution at stator outlet and the outlet static pressure is again a given function of radius. To fulfill correct nondimensionalization, these two ratios: ^^ and ^^ at the rotor inlet have to
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be prescribed, i.e.
Relative Mach number at points on the inlet is evaluated each step of marching procedure and W components are then computed using common isentropic relations. The results plotted in the same form like for the stator are in Fig.3 and Fig.4. This model does not take into account the influence of body forces, i.e. Coriolis and centrifugal forces. The next approach keeps the above mentioned formulation and adds body forces to show how much do they influence the flow field structure. Fig.5 shows the results, achieved by method (b), of the rotor flow with body forces in the form of inlet and outlet relative Mach number distribution (xaxis corresponds to radius and y-axis to relative Mach number). Considerable changes close to hub can be seen. The flow in hub section accelerates strongly in case there are no body forces (see also relative Mach number isolines in Fig.6.a), whereas in the case there are body forces the acceleration rate almost disappears (see isolines in Fig.G.b). The presence of body forces has only minor
Fig.3.a: pressure side, method (a)
Fig.S.b: pressure side, method (b)
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influence at tip part of the cascade.
Fig.4.a: suction side, method (a)
Fig.4.b: suction side, method (b)
Fig.5.a: without body forces
Fig.5.b: with body forces
Fig.6.a: without body forces
Fig.6.b: with body forces
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Bibliography
[ANG 96]
ANGOT P., FURST J., KOZEL K.: TVD and ENO Schemes for Multidimensional Steady an Unsteady Flows - A Comparative Analysis, Proc. Finite Volumes for Complex Applications - Problems and Perspectives, editors: Benkhaldoun F., Vilsmeier R., Rouen 1996, Hermes, pp 283-290
[CAU 89]
CAUSON D.M.: High Resolution Finite Volume Schemes and Computational Aerodynamics, Notes on Numerical Fluid Mechanics Vol.24, Braunschweig 1989, Vieweg, pp 63-74
[FOR 94]
FORT J., HUNEK M., LAIN J., KOZEL K., SEJNA M., VAVRINCOVA M.: Numerical Simulation of Steady and Unsteady Flows Through Plane Cascades, Proc. 14th ICNMFD, editors: Desphane S.M., Desai S.S., Narasimha R., Bangalore 1994, Lecture Notes in Physics, Springer, pp 461-465
[FOJ 94]
FORT J., HULEK T., KOZEL K., VAVRINCOVA M.: Numerical Simulation of 2D Steady and Unsteady Transonic Flows, Transport Theory and Statistical Physics, 23(1-3), 1994, pp 385-407 editors: Desphane S.M., Desai S.S., Narasimha R., Bangalore 1994, Lecture Notes in Physics, Springer, pp 461465
[HUL 92]
HULEK T., HUNEK M., KOZEL K.: Numerical Solutions of Euler and Navier-Stokes Equations, Proceedings of the European CFD Conference, Vol.1, ed. Ch. Hirsch, J. Periaux, W. Kordula, Elsevier Science Publisher D.V., Amsterdam 1992, pp 61-68
[NI 81]
Ni R. H.: A Multiple Grid Scheme for Solving Euler Equations, AIAA Journal, Vol.20, No.l, 1981
Acknowledgement This work is supported by the grants No. 101/98/1229, No. 201/99/0267 and 101/98/K001 of Grant Agency of Czech Republic and by the grant COST OC Fl.70/1997.
An efficient and universal numerical treatment of source terms in turbulence modelling. B. Merci, J. Steelant and E. Dick Department of Flow, Heat and Combustion Mechanics Universiteit Gent, Sint Pietersnieuwstraat 41, B-9000 Gent, Belgium. Tel. +/32/9/264.33.14 Fax+/32/9/264.35.S6 e-mail: [email protected] ABSTRACT A careful treatment is required for the source terms in turbulence models in order to obtain a stable discretization. Implicit or explicit treatment is possible. The analysis shows that the choice depends on the source terms' Jacobian, which can be constructed either exactly or approximately (on the basis of individual terms). Numerical results prove that the rigourous method is generally applicable and superior to the approximate method, concerning convergence speed. The reason is that the multigrid technique can be applied, which is not at all evident for turbulent flow. The analysis indicates the need for a time step restriction, too, for numerical robustness. Practical results show this is not always necessary. Different two-equation turbulence models are investigated confirming the generality of the approach. Key Words:
low-Reynolds turbulence models (k-e;k-uj), source terms, multigrid
1. Introduction Nowadays, low-Reynolds two-equations turbulence models are often used in CFD codes. They describe the physics of turbulence more universally than algebraic models and are less complex than Reynolds' stress models. When using these models, the source terms should be appropriately handled in order to obtain a stable discretization. A numerical analysis shows that the choice between implicit and explicit treatment depends on the Jacobian of the source terms. This Jacobian can be constructed either exactly or approximately on the basis of individual terms. Independently of the turbulence model, the latter method is mostly used. In this work, a more fundamental study is carried out to analyse in a general way differential equations with (non-linear) source terms. A comparison of convergence results with both methods is carried out. Different turbulence models are investigated in order to check the generality of the approach. Finally, convergence acceleration using the multigrid technique is studied. Previous studies [STE 94, STE 97] showed that the application of the multigrid technique on turbulence equations is not evident. Gerlinger et al. [GER 97] observed the same difficulties and proposed to freeze the non-linear parts in the source term. In this study, the MG-technique is applicable without damping or freezing any parts of the source terms, if the proposed discretization method is used. Convergence is much faster than can be obtained by solving the turbulence equations only on the finest grid.
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2. Treatment of source terms
2.1. Scalar equation 2.1.1. Linear source term In order to investigate the influence of discretization schemes for source terms in differential equations, a simplified equation is considered first with a linear source term:
where A is a real or complex constant. It is well known that A > 0 requires an explicit treatment, whereas an implicit treatment is best suited when A < 0. If A is complex, the same conclusions are valid, but now the sign of Re(A] is decisive. 2.7.2. Non-linear source term The clear distinction of discretization in the previous paragraph is a consequence of the fact that the source term S((f>) and |f always have the same sign. This does not hold any longer for a non-linear source term, making the choice of discretization less evident. To investigate this, eq. [1] is generalized to:
with 5 a non-linear function in 0. Depending on the source term £(), a non-zero steady solution can be obtained if a value 4>e ^ 0 exists for which S((f>e) = 0. The steady solution is assumed to be stable. In figure 1 a possible evolution of the source term 5(0) is given in function of . Analytically, 0e is reached in a monotonic way. Therefore it seems a reasonable requirement that the discretization always results in an update 6(j) in accordance with the source term sign:
There are two other requirements, more specifically related to turbulence modelling. The first demand is that (j> cannot become infinitely large, and the second one is that cannot become negative. An appropriate discretization that fulfills these three requirements has to be looked for. Implicit discretization of eq. [2] results in:
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Figure 1: Source term S(
Explicit discretization gives:
The amplification factors GE and GI are respectively:
For a prescribed amplification rate G, the necessary time step to obtain this rate is determined as:
Four different situations are possible (points A,B,C and D). Two characteristic examples are considered here. First point A (S > 0 and ff > 0) is considered. The explicit treatment [5] guarantees |G| > 1 independent of the time step. However, the solution n+1 —> oo for Ar -> oo which is not desirable, since a finite, stable steady solution is assumed. The best time step should result in a solution close to the exact one. If an approximation 0 can be estimated for the steady solution <J)e, the corresponding explicit time step can then be calculated from eq. [7] with G = $l$n. The implicit treatment [4] demands -^- > -j^ in order to assure |G| > 1. For $>n n+1 the upper limit of the timestep, the solution will also ^ —>• oo. Explicit treatment is the best choice for point A, with the time step calculated from eq. [7]. As second characteristic example, point B (5 > 0 and ff < 0) is studied. The situation does not differ much from point A, except that within a linear approximation, a 'natural' additional upper time limit can be proposed. With Taylor's expansion, the source term S is approximated as S((pn+1) = 5( n ) + ff n (<£ n+1 -
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want 5(0 n+1 ) = 0, this results in 0 n+1
= 0n - ^ffa
an explicit treatment [6] to a time step ^^- = - |4
*
, which corresponds for 0n
. The corresponding n+1
coincides with point B' in figure 1. If by a non-linear theory a better estimation 0 can be found, the needed time step should be given by eq. [7]. In most cases, however, some hypotheses have to be made to obtain this non-linear approximation. To ensure a stable iteration process, the maximum time step is taken as the smallest time step from the linear and the non-linear analysis:
An implicit treatment [4] always fulfills the condition \G\ > 1, and moreover guarantees \G\ < oo for all A£/. Therefore implicit treatment is preferable for point B. An infinite time step, however, corresponds to a solution (frn+1 — <^#/, which can be far away from
Point C (S < 0 and ff < 0) is very similar to point B. The only difference is that the source term is negative, so that the desired amplification 0 < G < I. The conclusions and time step restrictions are the same as in the previous case. For the same reason, the situation of point D corresponds to point A. Again, the difference is that, as S(<j>) < 0 the amplification should be 0 < G < 1. Thus it is seen that neither of the two possibilities (explicit or implicit treatment) can fulfill all the three mentioned requirements on itself in any of the four situations. The introduction of a time step restriction seems therefore necessary. For an explicit treatment this is obvious, since an infinite time step leads to an infinite (positive or negative) value of . For an implicit treatment, however, it can be necessary, too. In practice, however, it is difficult to approximate G. When ff < 0, implicit treatment is more robust than explicit treatment. In order not to counteract the implicit treatment, equations [8] and [9] are slightly altered. In [8], the sum is taken instead of the maximum. A closer look at eq.[7] reveals that this corresponds to dropping the term ff n
^ (j)
in [9]. Based on this discussion, the time step to be taken in any case is given by:
combined with an explicit treatment of the source term S((f)) if ff > 0, and an implicit one if ff < 0. It is clear that the practical use of equation [10] requires a reasonable approximation of G.
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2.2. Coupled system The source terms in two-equation turbulence models are in general strongly nonlinear and coupled. Two major methods exist to handle the source terms. In the first method [VAN 86, WIL 93], the positive parts of the source terms are not linearized and treated explicitly. The negative parts are linearized in such a way that negative real eigenvalues are obtained. They are treated implicitly. The second method [MER 93] starts with the construction of the exact Jacobian of the source terms: |4. Depending on its eigenvalues, this matrix can be split into a positive and a negative part: |^ = ^- + ^-. According to the previous analysis, the distinction between positive and negative here is made on the basis of the real part of the eigenvalues. The negative part is then treated implicitly, the positive part explicitly. An additional time step restriction is in principle necessary for robustness. For k-£ or k-u based turbulence models, this time step can be determined from a simplification of the turbulence equations ([MER 99]). Numerical results, however, show that for realistic flow situations this time step restriction is less stringent than time step restrictions coming from the convective or the diffusive terms. These restrictions were not encountered in the previous analysis because there the equation was simplified towards one containing a source term, but no convective or diffusive terms. 3. Numerical results Both source term discretization methods are numerically investigated. The lowReynolds k — e model by Yang-Shih and the low-Reynolds k — u> model by Wilcox are studied for incompressible flows. A multistage time stepping scheme is used to reach the steady state solution [VIE 98]. From now on, the method with the approximated Jacobian will be called 'approximate method', whereas the other method will be called 'rigourous method'. Fig. 2 shows the convergence evolution for a fully developed channel flow. The convective terms and the diffusive terms are set to zero, so that the source system's behaviour can be studied. Within the approximate method (curves 1 and 4), there is always a time step restriction, resulting from the implicit treatment of the negative parts of the source terms. For the rigourous method, a calculated time step restriction has to be introduced (curves 3 and 6), or strong underrelaxation is necessary, resulting in a worse convergence (curves 2 and 5). This shows the (theoretical) necessity of a time step restriction in order to retain a robust method. Fig. 3 shows the convergence history for a flat plate flow on a stretched grid (145x89 points). An alternating line solver is used. The approximate method performs equally well as the rigourous method (curves 1 and 2, and curves 4 and 5). The reason is that the convective and diffusive time steps are more restrictive than the source term time step so that they cover the influence of the source term discretization. For the same reason, the introduction of a calculated time step restriction is not necessary here (curves 3 and 6). Fig. 4 shows convergence results for the k — £ model for a flat plate flow (193x97 gridpoints) using multigrid. The cpu times for the Navier-Stokes equations and for the turbulence equations have to be added to obtain the global cpu time. A substantial im-
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provement is seen with the rigourous method when both the Navier-Stokes equations and the turbulence equations are solved on the coarse grids (curves 5 and 6). Curves 3 and 4 correspond to the Navier-Stokes equations solved with MG and the turbulence equations only on the finest grid. Curves 1 and 2 show the result for all the equations only solved on the finest grid. However, no convergence is obtained when the first method is used in combination with full multigrid (curves 7 and 8). This indicates the superiority of the second method. For the k — LJ model, the difference between the two methods is much smaller. Both methods converge with multigrid, again leading to a substantial improvement over single grid calculations (results not shown). Fig. 5 shows convergence results for the k — e model for a backward-facing step flow. Similar conclusions as for the flat plate can be drawn. However, now the first method also converges with multigrid. Similarly as mentioned in [GER 97], the flat plate flow seems more demanding for the multigrid technique. Similar results are obtained for the k — us model (fig. 6).
Figure 2: Convergence history for channel flow.( 1: k — e, approximated Jacobian; 2: k — s, exact Jac.; 3: k — e, exact Jac., with time step restriction; 4: k — u, appr. Jac.; 5: k — ui, exact Jac.; 6: k — u>, exact Jac., with time step restriction)
Figure 3: Convergence history for flat plate flow with an alternating line solver.(1-6: see above)
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Figure 4: Convergence acceleration with the multigrid technique for flat plate flow.(1,2: residual of NS, resp. turbulence, eqs., all solved single grid; 3,4: res. of NS, solved with MG (4 grids), and turb. eqs., solved single grid; 5,6: res. ofNS with MG (4 grids), and turb. eqs. with MG (4 grids), rigourous method; 7,8: res. of NS with MG (4 grids), and turb. eqs. with MG (4 grids), approx. method)
Figure 5: Convergence acceleration with the multigrid technique for BFS flow.(l-8: see above) 4. Conclusion A method of source term discretisation is presented that is robust and, most importantly, allows the use of the multigrid technique for convergence acceleration. The method is independent of the turbulence model.
5. Acknowledgements The first author is aspirant at the Flemish Science Foundation (F.W.O.). 6. References
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Figure 6: Convergence acceleration with the multigrid technique for BFS flow. (1see above) [GER 97]
Gerlinger P. and Bruggemann D.", Multigrid Convergence Acceleration for Turbulent Supersonic Flows, Int. J. for Numerical Methods in Fluids, volume 24,p. 1019-1035, 1997.
[MER 93]
Merkle C.L., Weiss J. and Venkateswaran S., Efficient Implementation of Turbulence Modeling in Computational Schemes, Proc. Second U.S. National Congress on Computational Mechanics Washington, D.C., August 1993, 1993.
[MER 99]
Merci B., Steelant J., Vierendeels J., Riemslagh K. and Dick E., Treatment of Source Terms and High Aspect Ratio Meshes in Turbulence Modelling, Proc. 14th AIAA CFD Conference, Norfolk City, June 1999, in press, 1999.
[STE 94]
Steelant J. and Dick E., A Multigrid Method for the Compressible Navier-Stokes Equations Coupled to the k -e Turbulence Equations, Int. Journal of Numerical Methods in Heat and Fluid Flow, volume 4(2), p.99-113, 1994.
[STE 97]
"Steelant J., Dick E. and Pattijn S., Analysis of Robust Multigrid Methods for Steady Viscous Low Mach Number Flows, Journal of Computational Physics, volume 136, p. 603-628, 1997.
[VAN 86]
Vandromme D. and Ha Minh H., About the Coupling of Turbulence Closure Models with Averaged Navier-Stokes Equations, Journal of Computational Physics, volume 65, p. 386-409,1986.
[VIE 98]
Vierendeels J., Riemslagh K. and Dick E., A Multigrid Semi-Implicit Line-Method for Viscous Incompressible and Low Mach Number Compressible Flows, Proc. of the 4th ECCOMAS Computational Fluid Dynamics Conference, Athens, John Wiley, p. 1220-1225, 1998.
[WIL 93]
Wilcox D.C., Turbulence Modeling for CFD, Griffin Printing, Glendale, California, 1993.
Comparison of numerical solvers for a multicomponent, turbulent flow
Emmanuelle Xeuxet C.E.M.I.F. Evry Alain Forestier C.E.A. Saclay Jean-Marc Herard E.D.F. Chatou
ABSTRACT This contribution's topic is the resolution by different numerical solvers of a multicomponent, compressive, turbulent flow. The unique associated Riemann Problem's solution is identified thanks to an entropy characterization. An exact Riemann solver is implemented and called by Godunov scheme. Some numerical simulations are introduced to exhibit a comparison between Godunov scheme, Vfroe-nc and Rusanov scheme. Key Words: Turbulent flow - Godunov scheme - Linearized solvers
1. Introduction
In this contribution is exhibited the resolution of the hyperbolic system which describes a compressive multicomponent turbulent flow. The model is written for a poly tropic isentropic gas. With compressive flow, Favre's average is used to select a mean flow and a turbulent one. In this work, we are interested by the one order closure model and particularly the coupling between turbulence and pressure. Reynold's tensor is described through the turbulent kinetic energy K of the mixture. The system is closed thanks to the K evolution equation. 2. A turbulence model to describe multicomponent flows The average variables describing the flow are : (p,pa,pu,K]
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The density of the mixture is noted by p. pa stands for the density of one of the fluid, with a the mass fraction of one component of the mixture. U stands for the velocity and K for the turbulence of the mixture. Setting W = (C,K), this system is conservative in C(p,pa,pU) variable, but not in K variable :
With all these assumptions we obtain the following first-order system in 2D :
2.1 From a 3D problem to the 1D Riemann Problem It is well known that finite volume upwinding schemes are efficient methods to solve such no linear system. The most natural finite volume method is the Godunov's method. It requires the exact solution W = (C, K) on the boundary <9Qj. The Riemann solution of the multidimensional (S) system is unknown. So we have to come down to the solution of the associated one-dimensional problem written in the normal direction of the interface. X We set by (PR) the associated Riemann Problem, and by W*( —, Wi, Wr) its solution :
Come back to the Godunov scheme, we introduce the notations : V(i) the neighboring cells of cell "i" (not including cell i) lij the interface measure : lij = |5fit- Pi £lj\ nij the unit normal vector from the i cell to the j cell.
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We set by ^^(C*, rijk) the normal numerical flux at the interface between Wj and Wk~.
And, for the nonconservative equation :
Kj is an average of the Riemann solution on d£lj 2.2. Mathematical analysis of the hyperbolic system The (e> n ) system is hyperbolic, nonstrictly. c' is the turbulent celerity in a two phase flow.
The eigenvalues are the following ones :
And, the associated five eigenvectors are linearly independent in R 5 . The first and fiveth characteristic fields are genuinely nonlinear at the sufficient condition that pressure is a convex function of the - specific volume : P
The treble characteristic field is linearly degenerated :
The solution consists in at most six constant states separated by shock waves, rarefaction waves or contact discontinuities. The rarefaction curves are :
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Shock curves are :
The contact discontinuities verify
2.3. Entropy characterization and uniqueness of the solution The generic form of the mathematical entropy is :
Our system (S) admits two supplementary conservative variables :
In keeping with the second thermodynamic principle, a 9? convex entropy is growing on a physical shock.
The equivalence between Lax inequalities and compressive shock is shown. But the equivalence between entropy shock and compressive shock is demonstrated for only the physical entropy S. T has no physical sense, because its growing on
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1-shock curve implies incompressive shock. The computation of the Riemann solution on the entropy shock curves ensure the uniqueness of the result. 3. Other numerical methods and preferences
The advantages of the exact solver are well known : positivity respect, entropy solution. But we have to balance these advantages by the fact that the method uses more CPU than linearized solver which doesn't require analytical computations. Then we introduce different schemes to analyze where a method is more or less efficient than an other. 3.1 A linearized solver : Vfroe-nc Vfroe scheme was introduced by Faille, Gallouet, Masella in 1996. It is based on a local resolution of a linearized Riemann problem. The numeric flux is defined, like Godunov scheme, by the physical flux computed at the interface solution of the linearized problem. An extension of this scheme was introduced by Buffard, Gallouet and Herard [BUF 98]. Vfroe-nc uses the nonconservative variables to preserve Riemann invariants through contact discontinuities. Thus we prefer the variables (P(pa),U) to (pa,pU). 1 YI + Y With Y = (a, un, ut, K, P), r — — and the linearized variable Y = —-, 2 P
we have to solve the linear system :
With the turbulent celerity c'2 = (jP -\
10 K}t we obtain the eigenvalues : y
Defining the intermediate states by a combination of the eigenvectors r,- :
The linearizedRiemann solution y*(Yj,y r ,0) is given by :
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The extension of Vfroe-nc scheme to nonconservative systems is given by :
The linearized solver generates intermediate states that may not respect physical positivity. For example in the mirror states for a double rarefaction wave Uni < 0, we have to set :
It is also possible that the linearized velocity un implies a bad choice of a. A big jump of a may generate, in the next time step, such a pressure that the deduced mass fraction would be out of the (0,1) definition set. 3.2. Rusanov scheme Rusanov scheme was introduced in 1961, it doesn't need any exact or approximative Riemann problem's resolution. In a multidimensional framework, we have the nonconservative extension of the scheme :
We introduce Sij depending on the A eigenvalues :
Rusanov scheme has been chosen for its respect of p, pa and K positivity, and because it preserves the mass fraction in the good definition set : 0 < a < 1.
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3.3. Numerical results We present, in the following page, a 2D test case of a onecomponent flow through a tube with obstacles. The inlet and outlet are free entrance, the others walls are reflecting sides. The air is coming from the left to the right side of the tube, with an initial velocity of 600 m.s~l. The initial conditions are : (p, u, v, a, K, P) - (1.1,600, 0,1,1000,100000) This case of a tube with obstacles reveals the advantages and deficiencies of the different methods. Vfroe-nc generates some negative turbulence in spite of the boundary positivity. With Rusanov method, positivity problems do not appears, but the K evaluation is very approximate.
4. Conclusion We conclude that in some extreme tests the cost of an exact solver is justified by the robustness required. With Godunov scheme the entropy solution is safely captured, without any positivity problem. It is not the case with the other linearized solvers which have shown their limits. We advocate to use successively the two schemes : the exact one for initialization and to deal with the boundary conditions, and a linearized one to go on with a reduced cost.
5. Bibliography [BUF 98]
BUFFARD T., GALLOUET T., HERARD J. M., A sequel to a rough Godunov scheme : application to real gases, Ecoles CEA-EDF-INRIA pp.363-382, 1998.
[GOD 96]
E. GODLEWSKI, P.A. RAVIART, "Numerical Approximation of Hyperbolic Systems of Conservation Laws", Applied Mathematical Sciences 118, ED. Springer, 1996.
[HER 94]
J.M.HERARD, A. FORESTIER, X. Louis, "A Non Stncly Hyperbolic System To Describe Compressible Turbulence"', Rapport E.D.F/D.E.R. HE-41/94/11A.
[LOU 95]
X. Louis, "Modelisation Numenque de la turbulence compressible", PhD Thesis, univ. Paris VI, 1995.
[RAN 92]
RANDALL J. LEVEQUE, "Numerical Methods for Conservation Laws", Birkhauser, 1992.
[SMO 83]
J. SMOLLER, "Shock waves and reaction-diffusion Springer-Verlag, 1983.
equations",
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Figure 1: Comparison of turbulence at the critical time where K < 0 by Vfroe
Parallel Overlapping Mesh Technique for Compressible Flows
Jacek Rokicki Zentrum fur Hochleistungsrechnen, TU Dresden, D-01062 Dresden, Germany Warsaw University of Technology, Nowowiejska 24, 00-665 Warsaw, Poland { jack | jmajewsk \ jzoltak }@meil.pw.edu.pi
Dimitris Drikakis Queen Mary and Westfield College, University of London, Mile End Road, London El 4NS, United Kingdom
Jerzy Majewski Warsaw University of Technology, Nowowiejska 24, 00-665 Warsaw, Poland
Jerzy Zoltak Aviation Institute, Al. Krakowska 110/114, Warsaw, Poland
ABSTRACT In the paper we present the Chimera overlapping mesh technique applied to the solution of compressible flow problems. The presented algorithm is particularly suitable for cases with large number of meshes overlapping in almost arbitrary manner (including multiple overlaps).
Key Words: Compressible flows, Overlapping meshes, Parallel computation
1. Introduction
The Chimera overlapping mesh technique (e.g. [HEN 94, ROF 99]) is employed here as an efficient tool for grid generation in complex geometries and as a method for paralelisation. This technique is based on the subdivision of the physical domain into overlapping subdomains. Subsequently, the system of flow equations is solved on each subdomain separately (see Fig. 1). The global solution is obtained by iteratively adjusting the boundary conditions on each subdomain.
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Figure 1: Grid system Bl around the SKF-1.1 airfoil. 2. Implicit unfactored solver On each mesh the compressible flow is simulated by solving the Euler equations. Implicit time discretization is used and the nonlinear system of equations is solved by an unfactored Newton-type method. The Riemann solver [DRD 94] calculates the inviscid fluxes at the cell faces using the characteristic values of the conservative variables. A Godunov-type upwind scheme up to third-order accuracy has been employed for the calculation of the characteristic cell face values.
3. The overlapping-mesh algorithm We consider here the general case in which the flow domain ft is fully covered by several overlapping subdomains fti,..., ft/<-, ft = fti U ... U ft^. If F and Tj denote the boundaries of ft and ftj subdomains, respectively, then 7^ — Tj \ F will denote the interface between the subdomains in which boundary information takes place. In order to obtain convergence of the global solution, each interfacial point must be well covered by at least one of other subdomains. Each subdomain ftj is covered by a suitable regular grid. However, the grid points in the overlap region do not necessarily coincide (Fig.l). As a result
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each interfacial grid point has to be localised in the reference frame of each grid in the system. Subsequently biquadratic interpolation procedure is used in order to prolongate the local solution onto the whole subdomain. The second difficulty is related to the fact that the solution, obtained on various subdomains, remains different in the multiply overlapping regions. To deal with it weight functions XP are defined (henceforth called blending functions), which are used to construct the global solution (see [ROF 99] for details). 4 Global solution and full algorithm Let us assume that t/[p] is the solution vector defined on the subdomain fip and extended to the whole fi by assigning U[P](C) = 0 for C <£ Qp. The global solution U(C) is defined then as
The full algorithm encompasses the following steps: 1. Initialise boundary conditions at the interfaces jp (p = 1
, K).
2. Perform a few iterations using the Euler solver on each local grid to obtain new local solutions £7[p], (p — 1 , . . . , K). 3. Interpolate each local solution on the prescribed interfaces and calculate global solution. 4. Check the convergence on all interfaces and if there is no convergence return to step 2. The step 3 cannot be performed in parallel since it requires interprocessor communication, if different subdomains are served by different processors. Nevertheless, the computational cost associated with the step 3 is negligible compared to the cost of step 2.
5. Numerical results The numerical quality of the method was assessed using the case of the transonic flow around the supercritical SKF-1.1 aerofoil with the deflected maneuver-flap [STT 79]. The free-stream Mach number was M^ = 0.60 with the angle of attack a — 2°.
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Figure 2: Mach number distribution around the SKF-1.1 airfoil for MOO = 0.60, a = 2° (case B3). Three grid systems Bl, B2, and B3 with increasing degrees of refinement, were chosen to demonstrate the convergence of the numerical solution. The systems consisted of 20 grids (25 for B3) as shown in Fig. 1. The number of grid points was 18410 for Bl, 42183 for B2 and 55304 for B3. The Mach-number field corresponding to the grid system B3 is shown in Fig. 2. The shock wave is well visible in the middle part of the main airfoil, while a second, much weaker shock appears over the flap. Figure 3 presents the comparison of the computed Cp distributions on the main airfoil and on the flap for all grid systems used in the computations.
6. Parallel implementation Parallelisation of the method is based on the assignment of a single grid or a group of grids to each processor. Therefore, the decomposition of the flowfield into subdomains determines, at the grid generation stage, the granularity of the parallel problem. As a result, the number of processors L cannot exceed the total number of grids K. Moreover, since grids are of different size, the number of processors L must be significantly lower than K, otherwise an uneven
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Figure 3: Grid convergence study for the SKF-1.1 airfoil for M^ = 0.60, a — 2°. load distribution will severely affect the parallel performance. The parallel implementation scheme is shown in Fig. 4. The overall computational effort and total number of iterations did not depend on the number of processors. They were, however, slightly dependent on the number of grids in the system, overlap size, etc. (several results from numerical experiments can be found in [RDM 99]). To estimate the parallel speed-up we have measured the execution time TL necessary for performing a fixed number of iterations (in our case 100 iterations) on an L-processor system. The speed-up factor O.L and the corresponding efficiency r}L were defined as Q.L = TI/TL , T]L = &L/L . In order to quantify the load-balancing effects the coefficient (3 was defined by: ftp = Wp/j-^^q , ft — max p= i i ... i L ftp , where wp stands for the workload assigned to the p-ih processor (wp is proportional to the number of grid points assigned to the p — th processor). In order to minimise the coefficient ft, for a given system of grids, a simple grouping algorithm was used according to which grids (in descending load order) were assigned to processors with the lightest load. Various parallel tests were performed for the flow around the SKF 1.1 aerofoil using the 64-processor CRAY T3E. Two different grid systems were used. The first consisted of 25 grids with the total workload of 53344 (grid B3 from previous Section), with the largest and the smallest grids having workload of
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Figure 4: Parallel implementation scheme. 10000 and 35, respectively (allowing the use of up to 6 processors). Table 1: The acceleration factor OL total efficiency TH and load balancing coefficient /3 for a 64-processor CRAY T3E (grid system B3). OLL L TL [sec] f?L 0 1 — — — 1148 2 1.0004 1.96 585 0.98 0.97 3 396 1.0005 2.90 4 3.80 1.0011 303 0.95 0.94 5 1.028 4.69 245 4.72 6 1.125 243 0.78 Table 1 collects obtained the values of a^rji , /3. The case L = 6 is already poorly balanced and, therefore, no further acceleration is possible. For L = 5 the communication-to-computation performance ratio is visualised in Fig. 5, as obtained with the VAMPIR performance analysis tool [NAG 96]. The second grid system contained 57 grids allowing the use of a larger number of processors. The total workload was equal to 36747, and the workload of the largest grid was 1800. The parralel communication was organised either by PVM reduce or by PVM send/receive. The performance is presented in Fig. 6.
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Figure 5: Single iteration of the solver for CRAY-T3E using 5 processors communication to computation performance ratio. 6. Conclusions The present results reveal that the overlapping-mesh technique can be efficiently used on a multi-processor system in conjunction with implicit methods and characteristics-based schemes. Grid-independent solutions were achieved for the case considered here. The efficiency of such an approach will increase with the complexity of the problem, provided that the workload can be distributed between the processors as evenly as possible. This can be achieved by further subdivision of existing grids or by re-grouping grids assigned to processors. Acknowledgments This work has been supported by the KBN 7 T07A 022 14 and KBN 9 T12C 069 14 Grants. Performance tests were run on the CRAY T3E at ZHR, TU-Dresden. Bibliography [DRD 94]
DRIKAKIS, D., DURST, F., Investigation of Flux Formulae in Tran-
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Figure 6: The acceleration factor al, total efficiency rji and load balancing coefficient /3 for a 64-processor CRAY T3E (57 overlapping grids system). sonic Shock-Wave/Turbulent Boundary Layer Interaction, Int. Journal for Numerical Methods in Fluids, 18, p. 385-413, 1994. [HEN 94]
HENSHAW, W.D., A Fourth-Order Accurate Method for the Incompressible Navier-Stokes Equations on Overlapping Grids, J. Comput. Phys., 113, 1994.
[NAG 96]
NAGEL, W.E., ARNOLD A., WEBER M., HOPPE H-C., SOLCHENBACH K., VAMPIR: Visualization and Analysis of MPI Resources, SUPERCOMPUTER 63, 12, No. 1, p.69-80., 1996.
[RDM 99]
ROKICKI, J., DRIKAKIS, D., MAJEWSKI, J., ZOLTAK, J., Numerical and Parallel Performance of the Overlapping Mesh Technique Coupled with Implicit Riemann Solver, ZHR Internal Report, ZHR-IR-9901, 1999.
[ROF 99]
ROKICKI, J., FLORYAN, J.M., (1999) Unstructured Domain Decomposition Method for the Navier-Stokes Equations, Computers and Fluids, 28, p.87-120, 1999.
[STT 79]
STANEWSKY, E., THIBERT, J.J., Airfoil SKF-1.1 with Maneuver Flap, AGARD Report AR-138, 1979.
A comparison of Finite Volume and High-Order Finite Difference Schemes for the Solution of the Navier-Stokes and Euler Equations
M. Meinke, Th. Rister, R. Ewert Aerodynamisches Institut Willlnerstrafle zw. 5 und 7, 52062 Aachen, Germany Tel.: +49-241-804821, E-mail: [email protected]
ABSTRACT The properties of finite volume and high-order discretization schemes are compared first for the large-eddy simulation of turbulent flows and secondly for computational aeroacoustics. For the LES of turbulent flows a finite volume method of second-order accuracy based on a slightly modified AUSM formulation, [LS93, MSR98] is compared with a compact finite difference method following Lele [Lel92]. It is found that no advantage is obtained from the high-order finite difference scheme for a plane turbulent jet flow due to the fact that filtering of the solution is required to avoid high frequency oscillations. In the field of aeroacoustics the application of high-order methods is, however, indispensable. This is demonstrated with solution of the Euler equations for acoustic benchmark problems, which involve the simulation of the propagation and interaction of acoustic and entropy waves. For different cases the accuracy of a second-order finite-volume and high-order discretization methods are compared, a compact finite-difference scheme derived by Lele and the dispersion-relation-preserving (DRP) scheme of Tarn [TW93a], which minimizes errors by using optimized approximations of first derivatives in the wave number and frequency space. Key Words: High-order methods, Large-Eddy Simulation, Aeroacoustics.
1. Introduction The direct computation of sound generated by a turbulent flow field is difficult, since the fluid mechanical pressure variations are several orders of magnitude larger than the acoustic signals. One way to solve this problem is to extract
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information about the acoustic field from a flow simulation and to use this information as source terms in computational aeroacoustic (CAA) methods. The LES of turbulent flow field seems to be well suited for the determination of the source terms, see e.g. []. For the simulation of the sound propagation in a non-uniform flow field the Euler equations have to be solved. Discretization schemes for these two problems are considered in this paper. Schemes of higher than second-order are usually more difficult to formulate and less robust, but promise a more accurate solution with the same amount of grid points. This can be important in the field of LES, where the truncation error of the numerical scheme can have a substantial influence on the solution and interferes with the contribution of the subgrid scale model. High-order discretization schemes and a finite volume method of second-order are compared for the solution of the Navier-Stokes equation for the LES of a turbulent plane jet and for the solution of the Euler equation for the simulation of wave propagation. 2. Method of Solution
The LES of the turbulent flows was carried out with different solution schemes for the Navier-Stokes equations, a second-order finite volume method based on a AUSM method and a compact finite difference scheme of 6th-order accuracy in space. The discretization of the viscous fluxes is performed with central differences of second-order accuracy. Test simulations with a sixth-order scheme for the viscous terms in combination with the compact finite difference scheme did not provide more accurate results compared to those obtained with the second-order scheme. For the simulation of sound propagation the Euler equations are solved additionally with a fourth-order Dispersion Relation Preserving (DRP) scheme. In the following the discretization methods for the convective terms are described briefly. In all cases the Navier-Stokes or Euler equations are discretized in general curvilinear coordinates for collocated structured grids. 2.1 Second-order AUSM-Scheme A modified AUSM-scheme is used for the discretization of the convective fluxes with a second-order accuracy. The formulation for the convective flux F in direction /3 is as follows:
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Herein, p is the density, u is the velocity, p the pressure and e the internal energy. Superscript L and R denote right and left interpolated variables which are obtained with a quadratic MUSCL interpolation of the primitive flow variables. M/3 = ^- is the average of the left and right interpolated Mach number. The speed of sound a is also computed as an average of the left and right interpolated values. The parameter x is used to control the amount of numerical dissipation within this scheme. Its influence on the solution is demonstrated in [MSR98, Sch97] for the LES of a turbulent channel flow. In case x is set to zero, a central discretization of the pressure derivative is obtained, which exhibits only a very little numerical dissipation and was therefore found to be best suited for the LES of the different flow problems investigated. For strongly curvilinear or stretched grids a value of x=0 may result in an unstable scheme, so that slightly higher values must be used. For the CAA applications only values of x=0 were used. 2.2 High-Order Compact Finite Difference
Scheme
The high-order compact finite difference scheme proposed by Lele is implemented for the skew-symmetric formulation of the governing equations. The inner scheme of sixth-order accuracy is given by
Fp denotes the first derivative of the convective fluxes F@ at the grid point defined by the subscript p. Approaching the boundaries with non-periodic conditions the order of the scheme is reduced in steps from 6th to 4th and then to 3rd order. The reduction of accuracy near boundaries leads locally to larger errors and causes the generation of high-frequency oscillations. These oscillations could be avoided with a higher grid resolution in those regions. A substantial reduction of the spatial steps may be acceptable for direct numerical simulations, but not in a context of a LES. Therefore an explicit filtering procedure is applied every 5th time step with an one-parameter family of fourth-order accuracy:
Qp denotes the filtered variable at the node p. The parameter a allows a control of the filtering. Values only slightly smaller than a = 0.5 result in a cutoff filtering with a filter width equal to the grid spacing. Values of a in the range of [0.49-0.499] were used to damp high-frequency oscillations. All details of the present implementation are presented in [Ris98].
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2.3 Dispersion Relation Preserving Scheme The DRP-scheme following Tarn and Webb [TW93a] is used for the solution of the Euler equations. With the help of Fourier transformation in space and Laplace transformation in time the Euler equations can be recast into the dispersion relation. If the same procedure is applied to the discretized Euler equations and the dispersion error is minimized for a certain wave number range [TW93b], the coefficients aj of a finite difference approximation
can be determined with the restriction that a formal accuracy of 4th order is obtained. The coefficients then read: UQ — 0, a\— -o_i = 0.770882380518, 02 = -o_ 2 = -0.166705904415, a3 = -o_ 3 = 0.0208431427703. 2.4 Integration in Time The integration in time of the differential equations is carried out with an explicit Runge-Kutta method for both discretization schemes. For the secondorder scheme a low storage method, [Jam83], with second-order accuracy is used. A comparison of a classical Runge-Kutta method of 4th order and the low storage version for the high-order method showed that the same accuracy is obtained for the cases considered here, since the time step is relatively small due to stability restrictions. The computational effort for the integration of the same physical time is about four times higher for the high-order scheme than for the second-order scheme. This is due to the fact, that the computation of the space derivatives is about two times more expensive for the compact scheme. Another factor of two results from the smaller time step in the 4th order Runge-Kutta method, which can only be chosen half the size of that for the low storage version. Therefore, the low storage version was used in most of all cases with the compact scheme. 3. Results In the following results are presented from solutions of the Navier-Stokes equations for the LES of turbulent flows and from solutions of the Euler equations for CAA applications. 3.1 LES of a Turbulent Plane Jet It is known that jet inflow conditions have a large influence on the downstream development of the jet, [RS93]. In order to provide a physically mean-
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ingful inflow condition for the mass flow, a turbulent inflow condition is used, which is generated by the LES of a fully developed turbulent channel flow. Both, the jet and channel simulation, are conducted simultaneously, so that at each time level an instantaneous solution from a cross section of the channel flow can be inserted in the inflow plane of the turbulent plane jet. Plane Jet case 1 case 2
grid points 81 x 149 x 65 65 x 129 x 33
Lx x Ly x Lz | Ax1 | Ay1 10D x 8D x 2.5D 0.05D 0.005D 12.5D x 8D x 2.5D 0.15D 0.005D
Table 1: Grid parameters for the LES of the turbulent plane jet at a Reynolds number of 7600 and a Mach number of 0.1. x denotes the streamwise, z the spanwise direction. Ax1 and Ay1 are the smallest spatial steps in x- and ydirection, respectively. The fully developed turbulent channel flow is simulated for a Mach number of Ma=0.1. Quasi periodic boundary conditions are used in streamwise direction, in which pressure and temperature fluctuations and the mass flow rate are considered to be periodic. Periodic boundary conditions are used in spanwise direction. A detailed description of the results obtained for the fully developed channel and pipe flows with the applied algorithm can be found in [Sch97, MSR98]. The other boundary conditions correspond to the case of a turbulent jet exhausting from an opening in a wall of infinite extension into a fluid at rest. On the lateral boundaries the mass flux was assumed to be the same as for the first inner surface and the pressure was held constant. In the exit plane nonreflecting boundary conditions, proposed by Thompson, [Tho87], and Poinsot & Lele, [PL92], formulated for general curvilinear coordinates, [Ris98], are applied. This boundary condition allows to use a small domain of integration, thus leading to a reduced computational effort. In spanwise direction periodic conditions are applied.
Figure 1: Simulation of a plane turbulent jet at a Reynolds number of 7600 with different solution schemes. Left: decay of centerline velocity u c l /u c /(x—0), Right: Half width B of the jet. The turbulent plane jet flow is computed with both the second-order AUSM
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and the compact finite difference scheme. The corresponding grid and flow parameters are summarized in Table 1. The grid points of the channel flow mesh are continued identically into the mesh for the jet flow. The Reynolds number is 7600 based on the width D of the jet which corresponds to ReT=200 of the channel flow calculation. In Fig. 1 solutions of the second-order scheme are compared with those from the higher-order Pade3/4/6 scheme. For the compact finite difference scheme an explicit filtering with a=0.499 is used to remove high frequency oscillations in the flow field, which are generated by the reduced accuracy near boundaries. It can be seen that the same length of the potential core is predicted with case 1 and case 2 grids and also with the different discretization schemes. The decay of the centerline velocity is predicted with the same accuracy until the end of the domain of integration. The spreading rate is in good agreement with the theoretical value of 0.097 (Fig. 1).
Figure 2: Comparison of the turbulence intensities in streamwise direction cr(u"). Results of the AUSM scheme with different SGS-models. +: case 1, without model, x: case 2, without model, D: case 2, dynamic model. In Fig. 2 the turbulence intensities o~(u") at different locations near the inlet sections obtained with the AUSM scheme are presented. Only negligible differences between the calculation with and without SGS-model on the coarse grid (case 2) are visible. The turbulence intensity computed on the finer grid (case 1) differs from that on the coarser grid (case 2) especially in the core region. This is partly caused by the different solution of the channel flow, i.e. the inflow condition of the jet, and partly by the different resolution in the jet domain. The second moments in Fig. 3 of the Pade-3/4/6 and the AUSM scheme for case 1 and case 2 calculations again show only small differences for the cases with and without SGS-model. The turbulence intensities for the compact scheme on the coarse grid (case 2) are higher in the vicinity of the inlet section, because of small high-frequency oscillations generated by the decreased order of accuracy near the physical boundaries. The usual way to avoid such oscillations is by grid refinement near these boundaries. On the finer grid, which has a three times smaller spatial step in streamwise direction, these oscillations vanish totally.
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Figure 3: Comparison of the turbulence intensity in streamwise a(u"} and spanwise direction a(w"} at different locations near the inlet section: ; case 1, without model, PADE-3/4/6 with a = 0.499; : case 2, dynamic model, PADE-3/4/6 with a — 0.499; : case 2, without model, PADE3/4/6 with a = 0.499; : case 1, without model, AUSM; : case 2, without model, AUSM Further variations of numerical parameters were carried out, but are not presented here for brevity. For a higher Reynolds number of 22000, the same tendency as described above is observed. A detailed description of the numerical methods and all results can be found in [Ris98j.
3.1 Simulation of CAA Benchmark Problems To examine the properties of the different solution schemes for CAA applications, the Euler equations are solved for the test cases of category 3, problem 1 and 2, which were defined on the first CAA-workshop [HRT95]. In this case the propagation of density, pressure and velocity pulses with Gaussian shapes are simulated in a base flow with Mach number 0.5. The initial position of the entropy and vorticity pulse are selected in a way that they reach the outflow boundary simultaneously. In Fig. 4 the density distribution is shown for a dimensionless time of 60. The boundary conditions applied are the non-reflecting boundary condition
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Figure 4: Left: Density distribution for a dimensionless time T = 60. DRP scheme, Tam's boundary condition. Right: Comparison of the density profile on the symmetry line for the different schemes. following Thompson [Tho90] and the asymptotic conditions of Tam [TW93a]. In Fig. 4 the density profile is compared for the three different solution schemes at a dimensionless time of 60. The second-order method clearly shows the largest deviation from the analytic solution, which confirms that it has the largest dispersive and dissipative errors. The entropy and vorticity pulses also show a deviation from the circular shape, which indicates a stronger anisotropic error distribution compared to the high-order schemes. Both high-order methods show good agreement with the analytic solution, also for later time levels. For this case the computing time for the DRP scheme is about three times smaller than for the compact finite difference scheme.
VJi
Figure 5: Left: Density distribution for a dimensionless time T = 100. Pade scheme, Thompson's boundary condition. Right: Comparison of the density profile on the symmetry line for the different boundary conditions. The performance of the different boundary conditions is illustrated in Fig. 5 for the compact finite difference scheme. Although high frequency oscillation of small amplitude are visible with Tarn's boundary condition the deviation of the solution near the outflow boundary are much smaller. These results are in
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agreement with comparisons by other authors, see e.g. [HSM95].
4. Conclusion High-order schemes and a second-order finite volume schemes have been compared for two different applications, the LES of turbulent flows and computational aeroacoustics. In the case of the LES of a plane turbulent jet the applied compact finite difference scheme showed no advantage in terms of accuracy over a second-order method. The computational costs for the compact scheme are about a factor of four higher than for the finite-volume method. In both cases the solution was found to be more or less independent of the application of subgrid scale models. In solutions of the Euler equations for CAA applications, it is known that high-order schemes are advantageous. The simulation of two benchmark problems on equidistant grids showed that the DRP scheme of Tam and the compact finite difference scheme produce results of similar accuracy, where the DRP scheme is about three time less expensive than the compact scheme. Currently, these comparisons are extended also to stretched and curvilinear grids. First test simulations have shown that filtering techniques or damping terms have to be used to avoid high frequency oscillations in the solution. References [HRT95]
[HSM95]
[Jam83]
[Lel92] [LS93] [MSR98]
[PL92]
HARDIN J. C., RISTORCELLI J. R., AND TAM C. K. W., ICASE/LaRC workshop on benchmark problems in computational aeroacoustics (CAA). Technical Report CP-3300, NASA, Hampton, VA, May 1995. HIXON R., SHIH S.-H., AND MANKBADI R. R., Evaluation of boundary conditions for computational aeroacoustics. Technical Report 95-0160, AIAA, 1995. JAMESON A., Solution of the Euler equations for two-dimensional transonic flow by a multigrid method. Applied Math. and Comp., 13:327-355, 1983. LELE S. K., Compact finite difference schemes with spectral-like resolution. J. Comput. Phys., 103:16-42, 1992. Liou M. S. AND STEFFEN JR. CH. J. , A new flux splitting scheme. J. Comput. Phys., 107:23-39, 1993. MEINKE M., SCHULZ C., AND RISTER TH., LES of Spatially Developing Jets. In R. Friedrich and P. Bontoux, editors, Computation and Visualization of Three-Dimensional Vortical and Turbulent Flows. NNFM 64, pages 116-131. Vieweg Verlag, 1998. POINSOT T. J. AND LELE S. K., Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys., 101:104129, 1992.
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[Pop97]
POPE D. S., A viscous/acoustic splitting technique for aeolian tone prediction. In Second Computational Aeroacoustics (CAA) Workshop on Benchmark Problems. NASA CP 3352, 1997.
[Ris98]
RISTER TH., Grobstruktursimulation schwach kompressibler turbulenter Freistrahlen - ein Vergleich zweier Losungsansdtze. Diss., Aerodyn. Inst. RWTH Aachen, 1998. Russ S. AND STYKOWSKI P. J., Turbulent structure and entrainment in heated jets: The effect of initial conditions. Phys. Fluids, A 5(12):3216-3225, Dec. 1993. SCHULZ C., Grobstruktursimulation turbulenter Freistrahlen. Diss., Aerodyn. Inst. RWTH Aachen, 1997. THOMPSON K. W., Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys., 68(1): 1-24, 1987. THOMPSON K. W., Time dependent boundary conditions for hyperbolic systems ii. J. Comput. Phys., 89:439-461, 1990. TAM C. AND WEBB J., Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys., 107:262-281, 1993. TAM C.K.W. AND WEBB J.C., Direct computation of nonlinear acoustic pulses using high-order finite difference schemes. Technical Report 93-4325, AIAA, 1993.
[RS93]
[Sch97] [Tho87] [Tho90] [TW93a]
[TW93b]
Simulation of 3D turbulent flow through steam-turbine control valves
Agaphonov B. N.1), Goryachev V. D.2), Kolyvanov V. G.1), Ris V. V.3), Smirnov E. M.3), Zaitsev D. K.3) 1) Energotech Ltd., P.S. Malii ave. 52-26, St. Petersburg, 1971 JO Russia State Technical University, Tver, 170024 Russia 3) State Technical University, St. Petersburg, 195251 Russia, E-mail: [email protected] 2)
ABSTRACT The paper deals with three-dimensional and axisymmetric computations of turbulent steady-state gas flow through steam turbine control valves of complex geometry. Using a finite-volume second-order multi-block Navier-Stokes solver, parametric computations have been carried out for two valve models. The flow essential features are discussed, and pressure losses calculated are compared with experimental air-test data. Key Words: finite-volume method, 3D turbulent flow, steam turbine, control valve.
1. Introduction Control valve is one of the crucial parts of a steam turbine. Efficiency and reliability of the valve depend, to a great extent, on flow behavior in its flow passage, which has a complicated geometry. A typical valve flow passage includes an inlet tube, a partially blocked steam chest, an annular contractor, a conical diffuser and an outlet tube. As a rule, the center-line of the inlet tube is perpendicular to the axis of the main part of the valve body that produces 3D flow inside the valve. The main parameter affecting the flow and pressure losses is the valve lift. Today's achievements of CFD allow routine 3D computations to be performed for development of more efficient and cost-effective valve design. An example of application of 3D numerical simulation for getting more insights into the behavior of flow through a steam turbine valve is reported in [COF 96]. The present work describes the first experience of use of an advanced Navier-Stokes finite-volume multi-block solver (named as SINF code), for studying turbulent compressible gas flow in steam turbine valves designed within the Russian industry.
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2. Test cases and mathematical models Systematic computations of steady-state flow for different valve lifts have been performed to compare a control valve designed following [KOL 98] with a prototype manufactured for many years for high-power steam turbines. Computational examples are formulated accounting for the conditions adopted during the air test of two control valve models [KOL 98]. For both models, the seat diameter, D, is 70 mm. Data for three cases are presented below. The main geometrical and operating parameters adopted at the calculations are given in Table 1 (G* is the critical mass bulk). Table 1. I
II
III
0.3
0.21
0.3
Mass bulk, GIG*
0.485
0.344
0.485
Inlet-tube Mach number
0.127
0.090
0.127
Reference velocity, m/s
95.7
67.9
95.7
Case Lift, hID
Cases I, II correspond to the valve model with an improved passage (Figure 1). The valve lift, h, is 0.3D for Case I, that is the maximum designed value. For Case II, the lift is reduced to 0.21D. Despite Figure 1 covers only the major part of the valve passage, the flow through the valve has been calculated simultaneously with the flow in the inlet and outlet tubes dropped in the plot. Note also, that the grid shown on the valve boundary surfaces consists of lines connecting centers of finitevolume boundary faces, and of inherent boundaries of computational blocks.
D
Figure 1. Example of control valve passage geometry
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Case III is associated with the valve of an old construction, the passage of which is far from a perfect aerodynamic form. The shutoff element has a plate-like form that does not fix the position of 3D-flow separation, the seat has a hollow, and there are two backward steps in the diverging part of the valve passage. The valve lift is taken the same as for Case I. For description of 3D turbulent flow within the valve body, the Reynoldsaveraged Navier-Stokes equations are applied in combination with the highReynolds-number k - e model of turbulence and the wall-function technique [LAU74]. Since the standard k - £ model produces an excessive turbulence generation in the contractor region; a simple procedure of eddy viscosity limitation is used up to the downstream edge of the shutoff element. The limit is taken to be equal to the maximum eddy viscosity within the inlet tube. The latter is defined using relations typical for fully developed round tube flow. As well, calculations based on axisymmetric approximation have been carried out to analyze the contribution of 3D effects on pressure losses in the valve design adopted. To get an axisymmetric formulation of the problem, the inlet tube of the real valve is replaced with an upstream annulus.
3. Computational aspects The SINF (Subsonic and INcompressible Flows) code is based on the second-order finite-volume spatial discretization using the cell-centered variable arrangement and body-fitted grids. Cartesian velocity components are used as primary variables. Upwind second-order schemes are applied to compute convective parts of fluxes (the results presented have been obtained with the QUICK scheme [LEO 79]). The artificial compressibility approach and implicit schemes incorporating the approximatefactorization and convergence-acceleration techniques are employed to solve both incompressible and subsonic compressible flows [SMI 93]. In the recent multi-block version, exchange of data at block interfaces is carried out at each artificial time step via creating an auxiliary virtual interface block, that involves geometrical and flow-filed information from near-interface layers of connecting grids. Computations for the virtual-block cells are performed with the same rules as for real blocks. This technique provides for full conservation of convective and diffusion fluxes even on non-uniform and strongly skewed grids. The 3D computations have been performed assuming the flow to be symmetrical with respect to the middle plane. For each valve model, a multi-block grid of 80,000 cells was generated. For that, as well for data post-processing, possibilities of the SELIGER system [GOR 97] are used. As an important part, this system includes an advanced grid-generation and geometry-modeling tool denoted ORIGGIN (Operative and Rational Interactive Grid Generation INstrument). 2D/3D block-structured grid generation possibilities are provided by implementation of various algebraic techniques and an elliptic equation solver. Another part of the SELIGER system is the graphic postprocessor LEONARDO.
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4. Results and discussion 4.1. Axisymmetric computations Figure 2 shows results of axisymmetric computations conducted for Case I. A distribution of velocity vector normalized with the reference value is given together with isolines of non-dimensional total pressure. One can see that a relatively small separation region is formed past the shutoff element, and the velocity defect is retained until the outlet section of the downstream diffuser. Note that the maximum value of the Mach number achieves 0.6. No flow separation is observed on the outer wall of the passage, but a wake-type layer developing along the outer wall gives a very significant contribution to the integrated pressure loss. The development of this, relatively thick, layer starts after a steep turn of the flow at the beginning of the diffuser. As a whole, the results of simulation point that the "eddy" part of pressure losses is dominant. That, in particular, promotes a successful use of the standard k - e turbulence model that works well at predictions of free shear layers. For the flow region shown in Figure 2, the pressure loss calculated is 1.52%. However, available experimental data were obtained with a rig where a tube five times diameter had been placed downstream the diffuser and a small backward step had been arranged in the connection plane. Simulation of axisymmetric flow developing in the region downstream the diffuser has resulted in a pressure loss of 0.54%. The overall rate of pressure loss is given in Table 2.
(a)
Figure 2. Velocity vectors (a) and total pressure contours (b) computed with axisymmetric formulation for Case I A reduction of the lift (Case II) leads to an increase of the separation zone past the shutoff element. Moreover, a higher non-uniformity of velocity field within the diffuser is observed. That would result in an increase of pressure losses. However, the reduction of the Mach number adopted for this case compensates the effects of
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flow field transformation, and the overall pressure losses have retained at the same level as in Case I (Table 2). Table 2. Case
I
II
III
Pressure loss, AP/Pinlet, % - air test data - 3D computation - axisymmetric approximation
2.0 2.3 2.1
2.0 2.3 2.1
3.8 4.0
Flow configuration computed for Case III (Figure 3) differs considerably, as compared with that for Case I. The geometry of the diffuser including two backward steps may come under obvious criticism. The overall rate of pressure losses computed has risen to 4%.
(a)
Figure 3. Velocity vectors (a) and total pressure contours (b) computed with axisymmetric formulation for Case III
4.2. Three-dimensional computations Flow structure computed on the base of 3D formulation is illustrated below for Case I only. Figure 4 shows distributions of the Mach number and total pressure over the symmetry plane. Note, that the gas inflows into the steam chest from the right. Comparing Figure 2 and Figure 4, one can extract the 3D effects. First, the adopted size of the steam chest is not enough to create a flow field that would be uniform in the circumferential direction. The velocity of gas in the gap between the seat and the shutoff element is higher on the inlet tube side than that on the opposite side. A significant circumferential non-uniformity of the flow field is
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observed even in the narrowest part of the flow passage Consequently, the shutoff element is acted upon by a non-zero lateral force. The separation zone is deformed, as compared with the axisymmetric data, and the low-speed wake is shifted to the right, i.e. in the direction opposite to the intake velocity.
Figure 4. Mach number (a) and total pressure (b) contours over the symmetry plane. 3D computations, Case I To get more insights into the 3D-flow structure, let us consider velocity distributions over several representative cross sections identified in Figure 1. Data for two sections (A, B) given in Figure 5 illustrate the development of flow inside the steam chest. Over both the sections, the velocity field non-uniformity is retained almost same, despite the velocity level is considerably increased in the second section. The cross flow patterns around the central body look similar for the most part of its length. The main flow passes a small flute of the central body without a massive separation, so the influence of this flute on pressure losses is not significant. The flow swirl generated within the steam chest due to the lateral intake, in combination with the separation past the shutoff element, results in a formation of longitudinal vortex structure in the lower valve body. It is illustrated in Figure 6 covering velocity distributions over other two sections (C, D). The cross flow developing inside the diffuser is of rather high intensity. Remarkably, that the
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direction of cross circulation is opposite to that typical for flow in curved tubes. It is reasonable to suggest that the real vortex cores developing in both halves of the valve body to be in an interaction. That produces their periodical displacement in the cross direction, and leads to development of low-frequency oscillations being a possible reason of the valve vibration. To study numerically these effects one needs to conduct 3D unsteady simulation.
Figure 5. Cross velocity vectors and whole velocity isolines, (a, b) and (c, d) correspond to Sections A and B as given in Figure 1
0.25
(c)
Figure 6. Cross velocity vectors and whole velocity isolines, (a, b) and (c, d) correspond to Sections C and D as given in Figure 1
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Despite accounting for 3D effects changes the valve flow structure considerably; pressure loss data obtained with the 3D and axisymmetric formulations are practically the same. Table 2 shows also that the results of computations are in a good agreement with the air-test measurements.
Acknowledgments The development of the SELIGER system possibilities is partially supported by the Russian Foundation of Basic Research, grant 99-07-90103.
Bibliography [COF 96]
COFER J. I. et al., Advances in steam path technology, General Electric Power Generation Paper, GER-3713D, 1996.
[KOL 98]
KOLYVANOV V. G., AGAPHONOV B. N., Control valve for steam turbine, Russian patent, RU 21091143, Cl, 1998.
[LAU 74]
LAUNDER B. E., SPALDING D. B., The numerical computation of turbulent flows, Comput. Methods Appl. Mech. Eng., 1974, Vol. 3, N 1, p. 269-289.
[LEO 79]
LEONARD B. P., A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. Methods Appl. Mech. Eng., 1979, Vol. 19, p. 59-98.
[SMI 93]
SMIRNOV E. M., Numerical simulation of turbulent flow and energy loss in passages with strong curvature and rotation using a three-dimensional Navier-Stokes solver, Report on the "Research in Brussels'92" grant, VUB, 1993.
[GOR 97]
GORYACHEV V. D., SMIRNOV E. M., A computer and information system for computer fluid dynamics: SELIGER, Proc. of the 15th MACS World Congress on Scientific Computation, Modelling and Applied Mathematics, Berlin, Vol. 3, 1997, p. 47-53.
Adaptivity,
Tracking and Fitting
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An Adaptive Hybrid Object-Oriented Code for CFD-Applications Adhoc3D
U. Tremelf, H. BleeckeJ, G. Brenner§, G. Greinerf | NEC-CCRLE research laboratories, § Institute of Fluid Mechanics, University of Erlangen, t Institute of Computer Science, University of Erlangen
ABSTRACT This paper presents an object-oriented (OO) approach for CFDapplications with attention being paid to conserve high computational performance. This is demonstrated by implementing an OO-simulation environment for flow simulations on hybrid unstructured adaptive meshes using OO programming in C++ combined with a lean flow solver kernel in FORTRAN 77/90. Key Words: Object-Oriented Programming, High-Performance Computing, CFD.
1. Introduction In the field of numerical simulation object-oriented programming (OOP) is used only to a limited extent although the benefits like greater productivity, flexibility, maintainability and reusability ([GAM 95], [STR 98]) of this modern type of software engineering compared to subroutine-based procedural programming are generally accepted. However it is usually not used in the high-performance community since it is suspected of being the reason for severe performance losses (due to runtime- and memory-requirements), in particular on vector computers. In the present paper, an OO-approach is described for numerical simulations particularly in the field of CFD which allows to retain high computational performance using the above mentioned features of OOP advantageously. The basic idea is to use the OO paradigm wherever useful and to define clear and
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Finite volumes for complex applications
simple interfaces in order to encapsulate highly optimized modules. The possibilities of such an approach are demonstrated through the implementation of an OO-numerical flow simulation environment on hybrid unstructured adaptive meshes in C++. The solver object encapsulates a numerical method for the integration of the Euler and Navier-Stokes equations which has been realized in FORTRAN 77/90, interacting through well defined interfaces. This wrapping technique enables the use of modular, encapsulated and highly optimizable subroutines for numerical computations and improves maintainance and exchangeability. Parallel aspects, considered in the design phase, strongly influence the simulation environment components. A parallel implementation based on message passing will be realized in the near future. The main focus of the implemented system, described in the next section, is on complex meshing management, a solution based adaptation process and, closely linked to that, a surface reconstruction process. The solver object, including the encapsulated lean flow solver kernel interface, and the basic principles of the underlying flow physics are described in section 4. First results and performance measurements are presented in section 5, showing that a high computational performance could be achieved.
2. OO-Simulation Environment for CFD The basic components of a CFD simulation environment are well suited to be encapsulated into three high level objects: • A control-object (AdhocSD, see figure 1) managing the actions during runtime, i.e. controlling IO, solution and adaptation process. • A mesh-object describing the discretized computational domain, in this case a hybrid unstructured adaptive mesh, internally subdivided into other objects. • A solver-object solving the governing PDEs and computing the numerical solution depending on the given initial and boundary conditions. The methods used internally for the numerical computations are encapsulated within this object.
3. Mesh-Management The complex administration of the adaptive hybrid mesh is encapsulated in different objects. The underlying principle is a decomposition of the domain in piecewise disjoint and non-empty subdomains containing sets1 of elements 1
set.
Note that here the use of a set implies that every item occurs not more than once in this
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Figure 1: Adhoc3D-Shell of different type whereas the union of all subdomains is the whole mesh. Each element is uniquely contained in one subdomain. Each subdomain-boundary is completely covered through a set of boundary patches belonging to this subdomain. These boundary patches contain a set of faces. All faces in a patch have the same boundary type which indicates, e.g., a special boundary condition. Neighboring subdomains share a common boundary patch exactly, i.e. all faces in this patch must be shared between the subdomains. Therefore the mesh itself contains only a set of all subdomain objects and a set of all boundary patches. This decomposition is necessary to support the message passing parallel programming model. Such a general description of the mesh and its subdomains can be expressed in OOP quite well using class hierarchies with abstract and derived concrete types. A subdomain is defined as an abstract type, i.e. that a subdomain class only defines a common interface and common properties of subdomains in general. For example, each subdomain must have a unique ID and should read itself from or write itself to a file. The mesh, using these methods, does not take care about the concrete type of subdomain behind this object. If the method is common to all subdomains (like asking for the unique ID) it will be implemented in the subdomain class itself. Otherwise the method is defined as a virtual function which must be implemented in derived classes defining concrete types. The correct implementation is then looked up at runtime. A concrete type of a subdomain is, for example, an unstructured region containing elements of different type or a structured block containing only one type of elements connected in a structured manner. In figure 1 these kinds of concrete types are shown. The use of common boundary patch objects enables the transfer of information like marker values, flags etc. over subdomain boundaries and also between different subdomain types. New subdomain types
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Finite volumes for complex applications
(Cartesian subgrids, . . . ) can be added in an elegant way, using this polymorphism.
3.1. Unstructured Regions The first concrete sub domain type currently implemented is an unstructured region. This object encapsulates sets of elements of different type connected in an unstructured manner. Supported elements are tetrahedra, pyramids, prisms and hexahedra. These elements have common faces like triangles and quadrilaterals which are the face types supported. The main design aspect was to realize compact and efficient objects for the management of all elements of one type in a certain subdomain. To achieve this the classes are designed like pools containing arrays of elements and not as arrays of fine grained objects. This means that these classes act like wrappers for internal, partly low level structures (one dimensional arrays). These arrays are used without duplication or adjustment in the solver low level kernel routines. This reuse of memory and data is an important aspect if only limited memory resources are available. It also helps to increase performance because no additional work is done for adjusting or translating these data. The internal arrays are totally encapsulated within these classes, other objects have only access to the internal data through well defined higher level methods. The elements and faces are not considered as independent objects. All data are managed in a pool. Each unstructured region contains a pool object for each element and face type. Through the use of generic programming these different element and face types are easily implemented. On top of the hierarchy there is a class containing common data used from all these pools. From this class a template is derived which contains all properties common to all elements and faces used here. These properties are, for example, the number of nodes, temporary marker data, links to parent and children elements. More special templates, namely the element-pool and face-pool templates are derived from this template. These pool templates contain the common properties of elements and of faces respectively. Concrete classes for the different element and face types are derived from these pool templates.
3.2. Adaptivity A local tetrahedral mesh adaptation facility has been implemented in order to accurately resolve flow field features. This facility is based on a combination of red (regular) and green (irregular) refinement, first proposed by R.E. Bank et al. in [BANK 83]. It is based on the idea that all elements marked for refinement are refined in a regular way first and then to correct the hanging nodes by applying local irregular refinement rules to the elements concerned. Irregular refined elements will first be regularly refined if further refinement is
Adaptivity, tracking and fitting
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Figure 2: Refinement patterns necessary. The algorithm used for the local refinement of tetrahedra is based on the approach of Bey in [BEY 95] and extended to create a stable and consistent triangulation 2 for decomposed meshes. This corresponds to a synchronized parallel closure of the mesh, currently sequentially simulated. Note that the local refinement could also be reverted which will be done automatically if elements are marked for coarsening. The possible refinement patterns, currently implemented for a tetrahedron and the corresponding refinement patterns for a triangle are shown in figure 2. If a triangle is regularly refined, the edge midpoints are connected resulting into four child triangles of equal size congruent to the parent triangle. An edge midpoint is connected with the opposite node resulting into two triangles, in case of an irregular triangle refinement. For a tetrahedron the regular refinement consists of cutting away four congruent child tetrahedra at the corners. The remaining octahedron is divided along the shortest diagonal into four tetrahedra. This regular refinement leads to eight child tetrahedra of equal volume. The irregular refinement patterns of a tetrahedron describe a split at one edge, at two opposite edges and at a face. Results of the adaptation process are shown in figure 4.
3.3. Surface
Reconstruction
Surface reconstruction is necessary in order to correctly represent the surface contour for newly inserted mesh nodes during the refinement process. The implemented procedure corrects the interpolated boundary nodes so that the original boundary is reconstructed up to a certain tolerance. Each boundary face is supplemented with a set of original surface points (so called progressive mesh information] which is used to reconstruct the surface if this boundary face is refined. Details of the geometric operations for this adaptation post2
In the sense defined in [BEY 95].
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processing step can be found in [TRE 99]. Some pictures to illustrate the procedure are given here (see figure 3).
Figure 3: Surface reconstruction through progressive mesh information
4. Solver Currently a solver object is available for inviscid and laminar flow simulations on adaptive hybrid meshes consisting of tetrahedra, pyramids, prisms and hexahedra using the dual-mesh idea combined with an edge-based data structure. This object takes care of extracting the necessary information needed in the kernel routines and sets up the structures used in the interface, without producing redundant data, i.e. • setting the coordinates, variables, connectivity lists and adaptation marker arrays for the tetrahedra (no copy, only a start address is given), • creating arrays containing the boundary nodes with the corresponding boundary type and computing the boundary normals, • creating an array containing all edges and computing the normals and control volumes belonging to each edge and node respective. After setting up these structures, the FORTRAN 77/90 flow solver kernel is called which performs the main computational work.
4.1. Flow Solver Kernel The fluid motion is described by the time-dependent laminar Navier-Stokes or the Euler equations in conservative form for compressible perfect gases. A box-type approach is used by constructing a secondary or dual-mesh defined by the centroids of the primary grid cells and grid edges as vertices. This secondary mesh defines the vertex-centered control volumes.3 The edge-based contribution to the cell residual at a given node is obtained using a modification of the advection upstream splitting method of Liou and Steffen [LIOU 93, WADA 97] 3
ing
This approach may be interpreted as a Finite-Element Galerkin method with mass lump-
Adaptivity, tracking and fitting 759 for the inviscid terms and central differences for the viscous terms. Second order accuracy of the original first order upwind scheme is achieved using the MUSCL approach with flux limitation. The gradients of primitive variables associated to the vertices of the primary grid are obtained employing the Green-Gauss formula. The resulting algebraic system of equations is integrated using a 4th other Runge-Kutta scheme. Convergence acceleration is achieved by local time-stepping. Presently, boundary conditions may be imposed for supersonic or subsonic in- and outflows based on the theory of characteristics. No slip wall boundary conditions are prescribed for viscous flows. Symmetry or a weak formulation for the "impermeable wall" condition is prescribed for the case of inviscid flows.
4.2. Adaptation Sensor The adaptation process is invoked depending on user settings and starts with a scan of the flow field based on a simple pressure sensor. The absolute pressure difference for each edge is scattered to the nodes. The sensor returns a list of markers for the tetrahedral elements, depending on the sensor values at the nodes and user settings.
5. Results and Performance Measurements First results are calculated on a HP-V-Class architecture, using the supersonic flow past a wedge (M^ = 3.0) with and without mesh adaptation. A fully converged steady-state solution was achieved in 800 iterations, including 5 refinement levels. The starting mesh has 2,599 nodes and about 12,230 tetrahedral elements. After five refinement levels, the mesh has 277,949 nodes and 1,665,685 elements. The overal computation time requires 12,410 secs (wall clock; without IO), of which the flow solver kernel takes 11,979 sees (96.5%), the flow solver wrapper 113 secs (0.9%) and the adaptation process 318 secs (2.6%). On the finest refinement level, the total memory requirement is 739 MB, of which the flow solver kernel takes 254 MB and the wrapper 60 MB.
Figure 4: Supersonic flow past a wedge without and with mesh adaptation (for each case mesh and contour is given)
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Portability of this method across various platforms is proven through different tests on machines with RISC architecture like SGI, SUN and HP. A port for both vector-parallel machines (the NEC-SX4) and massive parallel machines (the NEC-Cenju3- and NEC-Cenju4-system) is in progress. After finishing the parallelization of the current system, the integration of an industrial structured solver is planned in combination with dynamic load balancing through the use of the DRAMA-library [DRAMA 98]. The latter is necessary to avoid load-imbalance of the processors through the adaptation of the mesh.
6. Bibliography [STR 98]
STROUSTRUP B., The C++ Programming Language, 3rd Edition, Addison Wesley 1998.
[GAM 95]
GAMMA, E. et a/., Design Patterns, Addison Wesley 1995.
[TRE 99]
TREMEL, U., Simulation kompressibler nichtviskoser. 3D-Stromungen auf unstrukturierten adaptiven hybriden Gittern. Diplomarbeit am IMMD9, Universitat Erlangen-Niirnberg, 1999.
[BANK 83]
BANK, R.E. et a/., Refinement algorithms and data structures for regular local mesh refinement Scientific computing (Stepleman, R., ed.), pp.3-17, IMACS / North Holland, Amsterdam, 1983.
[BEY 95]
BEY, J., Tetrahedral Grid Refinement. Computing 55, pp.355-378, Springer-Verlag, 1995.
[DRAMA 98]
LONSDALE, G. et a/., Dynamic Re-Allocation of Meshes for parallel FE-Appl. 4th ECCOMAS CFD Conference, Athens 7-11 Septem., 1998.
[LIOU 93]
Liou M.S., STEFFEN C. J. et al., A new Flux Splitting Scheme, Journal of Computational Physics, 107, 23-39, 1993.
[WADA 97]
WADA, Y., Liou M.S., An Accurate and Robust Flux Splitting Scheme for Shock and Contact Discontinuities, SI AM J. Sci. Comput., Vol. 18, No. 3, pp. 633-657, 1997.
Adaptive mesh refinement for single and two phase flow problems in porous media
Mario Ohlberger Institut fur Angewandte Mathematik Hermann-Herder sir. 10 79104 Freiburg (Germany)
ABSTRACT This paper is devoted to the study o/a posteriori error estimates and adaptive methods for a mixed finite element - finite volume approximation for single and two phase flow problems in porous media. In a first step we analyze a scalar nonlinear convection diffusion equation and give a rigorous a posteriori error estimate for the Cauchy problem. In a second step these results are used to implement an adaptive method for the systems of density driven flow and two phase flow in porous media. Key Words: A posteriori error estimates, convection diffusion equation, porous media, two phase flow, finite volume schemes, unstructured grids
1. Introduction As a generalized mathematical model problem for single and two phase flow problems in porous media let us look at the following nonlinear system of partial differential equations. Let f2 C Hd, d < 3 be a bounded domain and J — (0,T) H R a time interval. Let QT = fi x J denote the space-time domain. We now look for a Darcy velocity u, a pressure p and a concentration or saturation c, satisfying in DT the system
Here
b and / denote given functions, where D may be degenerate.
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The system is closed by imposing suitable initial and boundary conditions. Discretising the system we use an operator splitting technique, where the equations (1) and (2) are discretised via mixed finite elements, using RaviartThomas elements, and the transport equation (3) is discretised by a cell centered upwind finite volume scheme. This choice of the numerical scheme is adapted to physical situations, where the flow is convection dominated, which is for instant the case in density driven ground-water flow or in an imiscible two phase flow regime. For a simplified model problem of the system, given above, the convergence of the scheme was proved in [OHL 97]. In this work we are interested in a posteriori error estimates and adaptive methods. Due to the difficulties of the system, the first efforts in this direction are the analysis of mixed finite element methods and finite volume methods separately for the corresponding scalar partial differential equations. In Section 2 we define a finite volume scheme for scalar convection-diffusion problems and give rigorous a posteriori error estimates. These results are obtained by using Kruzkov or Kuznetsov type techniques, such that no regularity of the solution is proposed. In Section 3 we define a mixed finite element method for a scalar saddle point problem corresponding to the equations (1) and (2) and state an a posteriori result of Wohlmuth and Hoppe [WOH 99]. Combining the mixed finite element and finite volume method we finally design an heuristic adaption strategy for the full system (1) - (3) in Section 4. Finally in Section 5 adaptive numerical experiments are given, which underline the theoretical results.
2. A posteriori estimate for finite volume schemes Let us first look at the Cauchy problem for the scalar nonlinear convection diffusion equation and let c be an entropy solution of this equation if for all
for all the following conditions.
with
For the data of problem
we assume
Furthermore let A, B be defined such that A the step and be a partition of Let J be a size of J. Furthermore for each n regular triangulation of R . The joint edge of Tj and TJ will be denoted by
Adaptivity, tracking and fitting 763 Sji. The oriented set of edges £n is defined as En := {(j,l)\Sji is an edge of Tn and Cj > ct}, where Cj, cl denote piecewise constant approximations of the exact solution c on the elements Tj,Tl}. Let h^in := mhijg/n diam (Tj) and let Xj be the circumcenter of triangle Tj. Define the distance between the circumcenters of two neighbouring cells Tj,Ti G Tn as dji := \Xj - xi\ and let hji denote the maximum of the diameters of those cells. We assume that there exists an a > 0 such that we have for all hj : = diam (Tj) and for all j, I G I. For any j, I £ In and tn G H+ let g£ : H -> H be a C1 numerical flux, satisfying the following conditions for all w,v,w',v' G [/4, #].
and
where Lg is a given constant and HJI denotes the outer unit normal to Sji with respect to Tj. Now the upwind finite volume scheme for computing the approximate solutions to (4) is defined by
for all n G {0,..., N} and j, I G In. Here N ( j ) denotes the indices of the neighbouring cells of Tj. Given the discrete values of let us denote the approximate solution ch : Hd x [0,T] -> M by ch(x,t) := cj,if x G Tj, tn < t < tn+1. For the time step A£n we assume the following CFL-condition
where £ G (0,1) and Lg is the Lipschitz constant from (8). With this definitions let us now state the main theoretical result of this paper.
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Theorem 1 (A posteriori error estimate) Let Ch be the discrete solution defined in (10) and c the entropy solution of (4)- Furthermore let the Assumptions (5)-(9) and (11) be fulfilled. Then we have
Here 770, Tfa, f]c and rjd are defined as
where
are defined as and are conand stants independent of the data. Lu and LI denote the Lipschitz constants of u, b respectively. Proof: The proof of this estimate is based on the Kruzkov or Kuznetsov technique and can be found in [OHL 99]. u It can be shown, that the dependence of the right hand side of this estimate on /i, A£ and e is of order
3. A posteriori result for mixed finite elements With the notations of Section 2 let us now define a mixed finite element method for a scalar saddle point problem corresponding to (1), (2). Therefore let us define the spaces V = {u <E (L 2 (ft)) 2 | div u 6 L 2 (D)} and W = L2(tt) Furthermore for v, w e V, y> e W and s e L°°(fJ) let the bilinear forms A and B be defined as:
Adaptivity, tracking and fitting
765
Then we look, for a given function s E L°°(ffc), at solutions (u,p) € V xW of the weak saddle point problem with homogeneous Dirichlet boundary conditions:
To discretise these equations let us introduce the Raviart-Thomas subspaces
has continuous normal components over the edges of ' Then the mixed finite element solution (u/up/J G Vh x Wh is defined as
In the case where the right hand side 7 = 0 and d — 2 Wohlmuth and Hoppe obtained the following a posteriori result [WOH 99]. Theorem 2 Let (u,p), (uh,ph] be defined as in (13)-(16). Then there exist hmax > 0 and cu, Cu independent of the mesh size, such that for all h < hmax where
Here H0 denotes the L2 projektion onto the space of piecewise constants, tji denotes a tangent vector to Sji, [-]j denotes the jump across this edge and |||u|||V = Jfi a~ 1 (s)u • u + J^ a~l(s)div(u)div(u). The weighting factor Uji is given by 0.5, if Sji is an interior edge and I elsewhere, ctji denotes the mean value of the extrema of a(s) on Tj and TI (cf. [WOH 99] for details).
4. Adaptive method for single and two phase flow Combining the a posteriori results of the Sections 2 and 3 we now want to develop a heuristic adaptive scheme for the system (l)-(3). Therefore let us define the error indicator fj0.j, rjn.j on elements Tj 6 Th as
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where Q^ is defined as in Theorem 1, $j := fT $/\Tj\ and Kt,Kc,Kd are given constants. Furthermore let rju]j be defined as in Theorem 2. Then the adaptive algorithm for solving (l)-(3) is: (while adapt0 Forn = 0 to N do: 1. For given s let Wh be defined as the Ph solution of the mixed finite element scheme (15),(16). adapt 3. For given scheme:
calculate
with the finite volume
Here adaptQ adapts the initial grid using the error indicators TJQ.J and returns false if it has done, whereas adapt calculates the grid Tn+l from the given data: adapt for all calculate if coarse refine until
from the given data; and
return the resulting grid as Here 9 G (0,1) is a given value, which depends on the refinement rules of the mesh. The tolerances TOL 0 ,TOL C and TOLU depend on T, the number of elements of Tn and a fixed prescribed tolerance TOL (cf. [KRO 99] for a possible choice).
5. Numerical experiments As a first example we look at the following scalar nonlinear and degenerate parabolic equation for the saturation s on 0 = (0,1) x (0,0.5) C B2, which models imiscible two phase flow in porous media
Adaptivity, tracking and fitting
767
where u = (1,0), b(s) is the fractional flow rate and d(s] the capillary diffusion given by b(s) = X l ( s ) / ( X l ( s ) + A 2 (s)), D(s) = X2(s)b(s)p'c(s). The mobilities AI , A2 of the phases and the capillary pressure pc are given by Ai(s) = s 3 /(2)U 1 ),A 2 (s) = (l-s)3/H2 andpc(s) = -^(l-s)/s. Furthermore in this example we choose /^ = l,/i 2 = 3, $ = 0.2 and K = 0.2. As initial condition we choose s = 0 in £1 and as boundary conditions we impose s — 1 on {0} x (0,0.5) and homogeneous Neumann boundary conditions on all other boundaries. Figure 2 shows the result of the adaptive numerical scheme, whereas Figure 1 shows the numerical result in the case, where we choose pc — 0 (hyperbolic case). In this case the equation reduces to the so called "Buckley-Leverett" equation. Both calculations where done with the same threshold value for the grid refinement.
Figure 1. Adaptive grid and saturation for the hyperbolic problem at t=0.1 (right) and comparison of saturation profiles on a horizontal slice (left).
Figure 2. Adaptive grid and saturation for the degenerate parabolic problem (right) and comparison of saturation profiles on a horizontal slice (left).
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As a second example let us look at the Elder problem, which is an example of density driven flow in porous media, described by the full system (l)-(3). Here a closed aquifer, which is initially filled with fresh water, is bounded by a salt dome at the middle part of the upper boundary. Thus salt water enters the domain and is distributed in the aquifer. Thereby the density differences between salt and fresh water initiate a convection field, which dominates the flow (cf. [SEG 94] for details). Figure 3 shows the adaptive grid, the concentration distribution of the salt water and the velocity field after 4 and 8 years.
Figure 3. Adaptive grid, concentration distribution and velocity field of the Elder problem after 4 and 8 years.
6.
Bibliography
[WOH 99] B.I. WOHLMUTH AND R.H.W. HOPPE, A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements. Math. Comp., posted on May 19, 1999, PII: S 0025-5718(99)01125-4 (to appear in print). [KRO 99] D. KRONER AND M. OHLBERGER, A-posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions. To appear in Math. Comp., 1999. [OHL 97] M. OHLBERGER, Convergence of a Mixed Finite Element - Finite Volume Method for the Two Phase Flow in Porous Media. EastWest J. Numer. Math., 5:183-210, 1997. [OHL 99] M. OHLBERGER, A posteriori error estimates for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations. Preprint 3, Mathematische Fakultat, Freiburg, 1999. [SEG 94]
G. SEGOL, Classical Groundwater Simulations: Proving and Improving Numerical Models. PTR Prentice Hall, Englewood Cliffs, New Jersey, 1994.
Parallel solution of hyperbolic PDEs with space-time adaptivity
Per Lotstedt and Stefan Soderberg Information technology Dept. of Scientific Computing Uppsala University SE-75104 Uppsala, Sweden [email protected] and [email protected]
ABSTRACT An algorithm is developed for adaptive solution of hyperbolic partial differential equations on structured grids partitioned into blocks. The space steps and the time steps are chosen so that estimates of the local errors satisfy given tolerances. The equations fulfilled by the global error are integrated on a coarse grid. The method is implemented on a parallel computer with dynamic load balancing. The numerical examples are a scalar equation and a system of equations in two dimensions. Key Words: adaptivity, hyperbolic partial differential tion, parallel computation
equations, numerical solu-
1. Introduction Adaptive methods for numerical solution of hyperbolic partial differential equations (PDEs) have the advantage of saving memory and computation time by spending the efforts in areas of the computational domain where it is necessary for the accuracy of the solution. In other areas, fewer grid cells are needed and larger time steps can be taken. Furthermore, if the refinement and coarsening of space and time steps are based on estimates of the numerical discretization errors, then the global error in the solution can be estimated. An error tolerance given by the user controls the acceptable error level. We describe an adaptive method for structured grids and finite volume discretizations in space. The grid is refined and coarsened by halving or doubling
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the step size in a multiblock partitioning of the computational domain. All cells in a block are refined or coarsened. This simplifies the administration of the grids. At the block boundaries there may be jumps in the cell size. The time derivatives are approximated by a predictor-corrector method and an implicit scheme for problems with large discrepancies in wave speeds. The errors in the time discretization are estimated in the same manner as in solvers of ordinary differential equations. The chosen techniques are suitable for parallel computation using message passing. A load balancing algorithm dynamically redistributes the blocks on the processors in an implementation of the method. The numerical tests are made in two space dimensions with a scalar equation and a system of equations. Grid refinements in patches of structured grids with application to flow problems are found e.g. in [BEG 89] [FEL 99] [PAK 96] [WRO 97]. In these papers either the steady state problem is solved or there is no adaptivity in time. A parallel implementation of an adaptive solver is described in [DLR 94]. Our work differs from the above cited references in that the local errors in space and time are controlled and integrated to obtain an estimate of the global error in the solution.
2. Space adaptivity The computational domain in space is partitioned into a number of blocks of a fixed physical size. Each block is a topological rectangle in 2D and has a structured grid of cells. At the boundaries of a block there is one extra column of ghost cells to facilitate the coupling to the neighboring block. There are jumps in the step size of a factor 2 along the common boundary of certain blocks where there is a refinement of the grid. The average of the solution in a cell is updated by computing the fluxes through the cell faces in the usual finite volume manner. This discretization will be second order accurate at least on Cartesian grids. The flux through a cell face at a block boundary is calculated using the variables in the adjacent ghost cell. If the cell in the other block has a different size then the values in the ghost cells are determined by averaging for a coarse ghost cell and bilinear interpolation for fine ghost cells. The interpolation in the fine ghost cells only give first order accuracy in the approximation of the derivative. Still, second order accuracy is obtained in the interior of a block when the characteristics of the solution cross the boundary [GUS 75]. The truncation error in the space discretization is estimated in the following way. In every block, a coarse grid is created by doubling the grid size in both directions. Suppose that the approximation of the space derivatives is Rh(uh), where Uh is the solution on the fine grid. Then the leading term in the error on the fine grid, assuming second order accuracy, is given by
Adaptivity, tracking and fitting
771
where puh is the prolongation or the volume weighted average of Uh in the four cells corresponding to one coarse cell. The estimate TR is computed for groups of four cells in the fine grid. If max \TR \ exceeds a tolerance SR, then the grid is refined in a block. If max \TR\ < O.I£R for all groups of cells in a block, then the grid is coarsened in that block. Since we do not allow the jumps in the grid size to be greater than 2 at block boundaries, neighboring blocks may have to be refined after a change. The check of the spatial error is made in every time step. If TR is too large, then a new grid is generated and the new time level is computed with the new grid spacing. If the estimate is sufficiently small, then the grid in the block is coarsened in the next time step.
3. Time adaptivity The time derivatives are discretized by linear multistep methods. The advantage with a multistep method compared to a Runge-Kutta method is that there are no internal stages where data have to be sent between the blocks in a parallel implementation. Two second order schemes are considered. One method is the predictorcorrector pair Adams-Bashforth (AB) and Adams-Moulton (AM) and the other is the backward differentiation method (BDF), see [HNW 93]. The coefficients in the methods are adjusted to a possible change in the global time step A^ between two time levels i and i + I to retain second order. Both AM and BDF are implicit and A-stable [HNW 93] and the imaginary axis belongs to the stability region. This is a desirable property for solution of hyperbolic problems. In the first method, the solution at level n + 1 is predicted by the explicit AB. Then the solution to the AM scheme is obtained by fixed point iteration. With a sufficiently small time step, only a few iterative steps are necessary for convergence. Convergence of the iteration sometimes introduces a more severe restriction on the time step than the accuracy. The local truncation error due to the time discretization is estimated by comparing the two solutions at tn+\ computed by AB and AM using the same old data. The local errors in the solutions are proportional to
where u(t] is the analytical solution and CAB and CAM are weakly dependent on At n . Hence, the local error in is
The error-per-step rt is computed for every cell in the blocks. Then calculated such that
is
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Finite volumes for complex applications
If
then Otherwise, In the second method, the explicit second order BDF discretization is used as predictor. Then un+1 is computed with the implicit BDF method. A system of linear equations is solved by GMRES in each step as described in [HLR 99]. There is no restriction on A£n for stability in time or convergence of GMRES. Only accuracy requirements introduce bounds on the time steps. This discretization and solution method is suitable for systems of equations with a large span of wave speeds. The local truncation error is estimated by comparing the explicit and implicit BDF schemes as in (2) and(3). For a linear system of hyperbolic equations
the error e = u(tn) — un in the solution approximately satisfies a similar equation where e(x,y,t) interpolates the errors at the discrete points in the space-time grid and TR is defined in (1). This equation tells us how the truncation errors are spread to the errors in the solution. We control only the right hand side of (6), but we integrate (6) to obtain an estimate of the global error e at every time level. This equation is integrated simultaneously on the coarse grid with twice the time step. The cost is only 1/8 of the cost of integrating (5) on the fine grid.
4. Parallelization The partitioning of the grid into blocks simplifies the parallelization of the adaptive algorithm and is suitable for processors with limited cache memory. The neighbors of a block are the same from the beginning, but the grid size across the boundary may change. Each processor is the host of one or more blocks. Solution variables at the boundaries are needed in an adjacent block to update the ghost cells. If a block and its neighbor belong to the same processor, then data are moved in the memory of the processor. If they are located on separate processors, then data are sent over the internal network with MPI. For parallel efficiency, the load balance must change dynamically as the work per time step in a block increases with refinement or decreases with coarsening of the grid. In order to keep the cost for load balancing at a reasonable level, only entire blocks are distributed not individual cells. This reduces the size of the data set passed to the load balancing algorithm to the number of blocks. The algorithm used in the numerical experiments is the recursive spectral bisection method (RSB).
Adaptivity, tracking and fitting
773
5. Numerical results The algorithms are tested on two simple hyperbolic equations (5) on a square, where A = B = 1 in the scalar case and the linearized and symmetrized Euler equations
for the system of equations. Periodic boundary conditions are assumed at the outer boundary of the computational domain.
5.1. Scalar problem The equation (5) is solved with A = B = 1. Initially, u(x, y, 0) is a Gaussian pulse. It is advected along the diagonal. After leaving the domain in the upper right corner it returns in the lower left corner. The error tolerances are £t — £R — 5 • 10~2 and there are 8 blocks in both the x and the y directions. In Fig. 1, the initial grid and the grid after one period are displayed. They are almost identical.
Figure 1: The computational grid initially (left) and after one period (right).
Figure 2: Comparison along the diagonal of the square between estimated (dashed) and analytical (solid) local errors TR (left), rt for the BDF method (center) and rt for the AM method (right).
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In Fig. 2 the estimated local errors in time and space are compared to the leading terms in TR and rt computed analytically. The errors overlap almost everywhere. The estimated global error from (6) is compared to the exact global error in Fig. 3. They agree very well. The time step of the predictor-corrector scheme is here restricted by the convergence of the fixed point iteration.
Figure 3: Integrated global error along the diagonal of the square from (6) (dashed) is compared to the exact error (solid) for the BDF method (left) and the AM method (right).
5.2. System of equations The equation (5) with the coefficients in (7) is solved with c = 1 and the initial data m = 0,u 2 = 0,u 3 = Q.3exp(-a((x - 0.5)2 + (y - 0.5) 2 )),a = 300. The error tolerances are EI = £R = 5 • 10~2 and the domain is partitioned into 8x8 blocks. There is no analytical solution to compare with in this case. The solution in «3 at t = I is found to the left in Fig. 4. The estimated error in 113 at t = 1 is plotted to the right. It is of the order of the tolerances as could be expected.
Figure 4: Solution of (5) (left) and computed error from (6) (right).
Adaptivity, tracking and fitting
775
5.3. Parallel execution The adaptive algorithm is implemented in Fortran 90 using MPI for the message passing. The experiments are executed on a set of three Digital Alpha Servers with four EV5/300MHz processors each. The load balance is updated in every 25th time step. For the scalar problem, the distribution of the blocks on 4 processors is depicted in Fig. 5. Here, a redistribution of the blocks is necessary since the refinement moves from the center to the corner of the computational domain.
Figure 5: The distribution of blocks at t = 0 (left) and t = 0.25 (right) for the scalar equation. Let Tp be the clock time for solving the system of equations (7) between t = 0 and t = I on p processors. In Fig. 6 the fixed size speedup IT^/Tp and efficiency 2T2/pTp are shown as a function of p. The comparison is made for 2 processors since the code does not run in single processor mode.
Figure 6: Fixed size speedup (left) and efficiency number of processors.
(right) as a function of the
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6. Bibliography [EEC 89]
M. BERGER, P. COLELLA, «Local adaptive mesh refinement for shock hydrodynamics», J. Comput. Phys., 82 (1989), p. 64-84.
[DLR 94]
3. DE KEYSER, K. LUST, D. ROOSE, «Run-time balancing support for a parallel multiblock Euler/Navier-Stokes code with adaptive refinement on distributed memory computers», Parallel Comput., 20 (1994), p. 1069-1088.
[FEL 99]
L. PERM, P. LOTSTEDT, Blockwise adaptive grids with multigrid acceleration for compressible flow, AIAA J, 37 (1999), p. 121-123.
[GUS 75]
B. GUSTAFSSON, The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comput., 29 (1975), p. 396-406.
[HNW 93]
E. HAIRER, S. P. N0RSETT, G. WANNER, Solving ordinary differential equations, 2nd ed., Springer-Verlag, Berlin, 1993.
[HLR 99]
L. HEMMINGSSON-FRANDEN, P. LOTSTEDT, A. RAMAGE, S. SODERBERG, ^Implicit solution of hyperbolic equations with space-time adaptivity», in preparation.
[PAK 96]
N. G. PANTELELIS, A. E. KANARACHOS, «The parallel block adaptive multigrid method for the implicit solution of the Euler equations», Int. J. Numer. Meth. Fluids, 22 (1996), p. 411-428.
[WRO 97]
J. Wu, H. RITZDORF, K. OOSTERLEE, B. STECKEL, A. SCHULLER, ^Adaptive parallel multigrid solution of 2D incompressible Navier-Stokes equations», Int. J. Num. Meth. Fluids, 24 (1997), p. 875-892.
Dynamic mesh generation with grid quality preserving methods
Andreas Wick and Frank Thiele Hermann-Fottinger-Institut fur Stromungsmechanik Technische Universztdt Berlin 10623 Berlin, Germany
ABSTRACT The various methods employed so far to accomodate computational grids to a moving boundary often fail to preserve grid quality. In this paper, a detailed analysis of this issue is conducted. Based on the findings a new method is presented that performs substantially better. The theoretical results are confirmed by numerical studies. Key Words: moving boundaries, deforming grids, dynamic meshes, spring analogy
1. Introduction The incorporation of moving boundaries and deforming domains has become an important task in numerical flow simulation. There are many applications in which this issue must be faced. To give a few examples, we can mention aerodynamic design optimization, fluid-structure interaction, free-surface flows and multi-element airfoils with moving flaps. In order to take into account the effects of time varying domains in an adequate and straightforward manner numerical algorithms are commonly based on the Arbitrary Lagrangian-Eulerian (ALE) formulation of the conservation laws (see e.g. [ANJ97]). There is no need for special search and interpolation algorithms as is the case with some other methods (e.g. [KAO95]). Compared to a standard finite-volume method that operates on a fixed grid (pure Eulerian approach) the numerical solution of the ALE equations additionally requires at each time step new grid point coordinates and the corresponding velocities with which the cell surfaces move.
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While the consistent and efficient computation of the grid velocities has been investigated thoroughly [ZHA93], the adjustment of the mesh to moving boundaries is still an open problem, at least for Navier-Stokes computations which require grids with very high aspect ratios. Specifically, for nontrivial geometries with boundaries exhibiting strong curvature or sharp bends, overlapping of mesh cells is likely to appear. In this case, a local or global remeshing becomes necessary and may induce unwanted interpolation errors. Therefore, of the methods proposed in literature, in section 2 we revisit shortly those having the potential to preserve the quality of the initial grid for the whole mesh evolution. Besides the well-known spring analogy model [BAT90], these are methods based on differential equations [ILI94, CRU96, LOE96]. The numerical implementation of the equations is addressed in section 3. In section 4 the methods are subjected to a deformation analysis which reveals serious weaknesses. The observed deficits also become apparent in numerical test cases. In section 5 the conclusions that can be drawn are summarized.
2. Grid generation methods
2.1 Problem formulation We assume that we start with a high quality grid at the first time level. Our aim is to find grids for the proceeding time levels that fit the prescribed boundary and that possess a quality comparable to the initial grid. Specifically, cross-sections of coordinate lines must not occur. To accomplish this, we have to inspect a very basic geometrical configuration, typical both for unstructured and structured grids. It is a line segment, connecting two neighbor vertices, which is subjected to a deformation. The difference between the coordinates at the new time level and the preceding time level defines the displacement vector.
We hope to find a displacement vector w that leaves the length of all line segments unaltered while transforming the grid of a former time level to a new one. Of course this is only possible if the prescribed boundary displacements correspond to a rigid body motion. If the boundary moves in another way, causing the interior domain to deform, the resulting stretches have to be equidistributed in a certain way. Otherwise a high local concentration of stretches will lead to a quick degradation of the grid quality.
Adaptivity, tracking and fitting 779 2.2 Methods A very well-known method is the spring analogy model. The lines connecting neighbor vertices are seen as tension and/or torsion springs. The imposed boundary displacement generates forces in the springs, leading to a new equilibrium state. By minimizing the energy of the spring system, the grid is optimally redistributed. A particular simple model that gave good results in many practical applications ([BAT90, FAR94]) has been proposed by Batina [BAT90].
V is the set of vertices and DST(f) the set of neighbors of i. In a structured 2D grid the set of neighbors N(z') consists of the east, west, south and north nodes. Despite some desirable properties of a numerical scheme derived from this equation, it has some flaws that gave rise to the development of alternative methods. Ilinca [ILI94] proposed to solve a Laplace equation for the grid velocity x. By an integration over the time step the displacement field is obtained. This numerical integration necessitates an assumption regarding the time dependence of the grid velocity (e.g. x = const, for tn < t < tn + Ai). The error introduced thereby leads to a time step restriction. It is therefore a better choice to use a Laplace equation for the displacement itself.
The introduction of a diffusion coefficient in this formula allows for more control over the grid evolution.
Crumpton and Giles [CRU96] choose the diffusion coefficient as a function of the cell volume. They argue that small cells are most endangered with regard to crossing of grid lines. Lohner and Yang [LOE96] prefer the diffusion coefficient to be connected to the wall distance, since large stretches are supposed to be close to the wall. While the cell volume is an intrinsic quantity of the discretisation, the wall distance is not, and as a consequence it might not be available. To keep the formulas valid for cases as general as possible we set
for the diffusion coefficient. The function Vol(xi) returns the volume of the dual grid cell which is associated with vertex #;. The Laplace equation [3] can also be interpreted as a special case (E = const., divergence free displacement field) of the following equation:
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This formula is also closely related to equation [2]. Similar to the derivation of the spring analogy model, the fact that the minimization of an appropriate energy function leads to optimally redistributed grids is exploited. But while the spring analogy is based an a discrete model now we apply a continous model. That is to say we consider the domain Q as a Hookean medium with zero Poisson's ratio. Then equation [6] represents the well-known Navier's equation of the classical linear theory of elasticity (see e.g. [ERI62]). Following the reasoning of Crumpton and Giles the Hook medium should be stiff for small cells. The most simple way to do this is to take the reciprocal value of the cell volume as elastic modulus E.
The coefficients of equations [4] and [6] are now strongly dependent on the chosen discretisation which might be considered as a disadvantage. A better choice should be to link the elastic modulus to the stretches whereby equation [6] becomes nonlinear. However, due to limited space we restrict ourselves to linear methods here.
3. Solution Procedure The discretisation of the differential grid generation equations are based on the Finite Volume Method (FVM). To allow complex geometries to be accounted for, a block-structured approach with general non-orthogonal coordinates is adopted here. The strong conservation form of the equations is retained by expressing the vectors and tensors in Cartesian components. The displacement vector is assigned to the vertices ("Cell vertex scheme"). First derivatives, appearing at cell faces, are approximated by central differences leading to a second order scheme. The numerical stability is enhanced by using the deferred correction approach which makes the scheme partly implicit. The system of linear equations obtained from the discretisation is solved by an iterative ILU method [STO68].
4. Theoretical and numerical analysis With regard to the grid quality the methods presented in section 2 have to meet two requirements. The main task is the equidistribution of stretches in a manner that avoids high local concentrations of them, since they are responsible for the overlapping of grid cells. Even more fundamental is the preservation of rigid-body displacements. It is evident that a method should not create stretches for boundary displacements that allow for a rigid displacement of the whole domain. It is important to note that these two basic requirements are sufficient to maintain a high grid quality, provided that the deflection of the
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boundary is small. This is normally the case as the movement of the boundary is subdivided in successively applied steps in order to resolve the time scales of the fluid flow. For small deformations there is a fundamental theorem which states that a general deformation can be considered as resulting from a linear combination of a rigid body translation, a rigid body rotation and stretches along the principal axes of strain ([ERI62]). If a method behaves well for these three types of deformations it is also able to deal with a general deformation since the latter is nothing but a superposition of the elementary ones. In the following section the equidistribution of stretches is further investigated.
4-1 Disc with a hole subjected to a rotated boundary The undeformed domain is covered by orthogonal grid lines. For the ratio of inner radius r;n and outer radius rout a value of 0.4 is chosen. The outer boundary is rotated in clockwise direction for an angle of a — 40°. Discretisations of the deformed disc with 20 x 10 grids are displayed in figure 1. For this
Figure 1. Deformation of a grid with constant spacing in radial direction by rotation of the outer boundary in clockwise direction; (a) Laplace equation, (b) Navier's equation with E = Vol(xi)~l, (c) spring model problem equations [3], [4] and [6] (provided that the diffusion coefficient and the elastic modulus are constant) have an analytical solution. It can be obtained by first transforming the equations from the cartesian coordinate system (x, y] to a polar coordinate system (
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As a measure of grid quality we use the angle between the grid lines after the deformation. It is obtained from the updated grid point coordinates. The angle is depicted in figure 2a and 2b as solid lines. The exact solution of equation [3] is very well represented by the numerical solution (circels in figure 2). We can therefore assume the discretisation to be fine enough. For the first test series a discretisation with constant cell volume V ( x i ] is selected. As can be seen in figure 2a, if the coefficient in the equations [4] and [6] are determined from the cell volume, the solutions coincidence with the analytical reference solution and hence lead to bad grid quality. While the discretisation with constant cell volume is rather academic, it is more natural to use constant spacing in radial direction. The same equations then give almost satisfactory results. While the Laplace equation leads to a worst angle of 16°, it is greater than 30° for Navier's as well as the diffusion equations, whereby the latter performs slightly better. It is important to note here that the distribution of stretches due to the coefficients [5] and [7] is further improved when the mesh is more refined towards the walls. For the two discretisations investigated, the spring model also preserves the grid quality in a satisfying manner, as can be seen in the figures Ic and 2.
Figure 2. Angle over normalized radius; (a) constant cell volume, (b) constant spacing in radial direction
4'2 Solid Body Rotation of a disc As already mentioned, it is important to examine the performance of the methods introduced in section 2 for the case of rigid-body deformations. To do so, the displacement vector that corresponds to a rigid-body deformation of the entire domain is inserted in the equations. In the case of rigid-body translation, the displacement vector is constant, which is indeed a solution of the equations [2], [3], [4] and [6]. Let us now take a closer look at the rigid-body rotation. The domain ^ shall rotate by an angle of a. The position vector X
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of the deformed domain is obtained by letting a rotation tensor a act on the position vector a; of the undeformed domain.
With
and
x we find for the displacement vector:
Here 6 denotes the identity tensor. Taking into account that yx = S, it is clear that the displacement vector as given by [10] is the solution of the Laplace equation [3]. This is not the case for the diffusion equation. If equation [10] is inserted it remains:
This equation can be considered as a linear, homogeneous system of equations which determines the components of the gradient V^- However, these components are already determined by equation [5]. In general this components will neither be zero nor will they have other values that fullfil equation [11]. To render equation [11] valid, the expression in brackets therefore has to vanish. For a Cartesian coordinate system this condition reads expanded:
This equation is not fulfilled unless the rotation angle is zero. Hence we conclude that equation [4] is not able to represent rigid-body rotation. Note that it makes no difference here if the coefficient k is determined from the deformed grid, as proposed by Crumpton and Giles. A similar analysis for the Navier's equation leads to: cos a = 1, which is approximately fulfilled for small rotation angles. We conclude that the Navier's equation is able to represent rigid-body rotation, provided that the rotation angle is small enough. It remains to examine the spring model. Inserting equation [10] into equation [2] we obtain:
The sum vanishes only if the whole domain is covered by straight coordinate lines, because in this case the contributions from opposite neighbor nodes cancel each other out. This is, of course, an unacceptable constraint on the spatial discretisation of complex geometries. Consequently, the first bracket is required to vanish which leads to the same condition as for the diffusion equation. Hence also the spring model cannot correctly represent a rigid-body rotation. The foregoing theoretical investigations are supported by numerical computations that are performed on a grid with constant spacing in radial direction,
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Figure 3. Grids generated for revolved boundaries; (a) diffusion equation with k = Vol(xi)~l, (b) Navier's equation with E = Vol(xi)~l, (c) spring model which was also used in the previous section. Both the inner and the outer radius are rotated in clockwise direction by an angle of 40°. The Laplace equation and also diffusion and Navier's equations reproduce the desired rigid-body rotation, provided the coefficients k and E are constant respectively. A slight deviation from orthogonality can be observed in figure 3b. This grid was generated with the Navier's equation with a variable coefficient k. For smaller angles this deviation disappeared. The use of the diffusion equation with variable coefficient led to a significant deviation from orthogonality (fig. 3a). Also the spring model fails to reproduce the rigid-body rotation, as can be seen in figure 3c. This weakness is probably the origin of the problems that were encountered by many authors (e.g. [RAU93]) when using the spring model. The degradation of the grid quality is expected to be larger the more the mesh lines deviate from straight lines in other words for strongly curved coordinate lines.
4-3 A complex test case After we investigated how the different grid generation methods perform for elementary deformations, we turn now to a general deformation. The geometry of this test case consists of a channel that contains a triangular shaped obstacle which fills one third of the channel high. In figure 4, magnifications of the mesh around the upper corner of the triangle are shown. They are generated by different methods in the following manner. First the triangle is rotated around its center in clockwise direction by an angle of 15°. Subsequently, the interior grid points of the initial grid are adjusted to the new boundary, which yields a new grid. Then the triangle is moved back to its original position again. From the grid produced in the latter stage and the known deflection of the triangle, a new grid is generated that fits the boundary of the initial state. It is clear that after this succession, the initial grid should be retrieved, or else the method employed to generate the grids is not quality preserving.
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The spring model fails completely, as it leads to overlapping mesh cells (figure 4c). In contrast, both diffusion equation and Navier's equation generate grids that can be used for fluid flow calculations. However, only the Navier's equation leads to a full recovery of the initial grid. When the diffusion equation is used the near wall orthogonality gets lost, as can be seen in figure 4a. This confirms the observations of the previous section.
Figure 4. Detail of the grid around the upper corner of the triangle after a succession of boundary displacements; (a) diffusion equation with k = Vol(xi]~l, (b) Navier's equation with E = Vol(xi)~l, (c) spring model
5. Conclusion Many authors observed that the methods employed so far for generating grids for complex time varying domains fail to preserve grid qualtity. By means of a deformation analysis, we were able to give an explanation for this. The spring model and the diffusion equation lack the ability to reproduce rigid body rotations. Problems will occur in applications where the boundary or a part of it rotates, e.g. with an oscillating airfoil. The Laplace equation, on the other hand, doesn't give satisfactory results in the presence of moving boundaries that exhibit strong curvature or sharp bends, as high concentrations of stretches are not equidistributed in a sufficient manner. On the basis of our findings, we proposed a new method that resolved these problems. This new method can be seen as a link between the other methods. It is important to note that the analysis conducted in this paper is valid in a very general setting. It does not rely on a particular coordiante system nor on the structured spacial discretisation. Hence, the drawn conclusions hold also for the threedimensional case and for unstructured grids. There are no significant differences with regard to the computational expense required by the different methods. However, in comparison with the computational expense of the fluid flow calculations, the overhead incurred by the grid generation is still too high. The implementation of more efficient solution procedures will be subject of further research.
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Acknowledgment The first author wishes to express his appreciation for the financial support to this work provided partly by the Dr. Fritz Walter Fischer-Stiftung and the DFG (grant No. AF 3/12-2).
Bibliography [ANJ97]
ANJU A. et al., 2-d fluid-structure interaction problems by an Arbitrary Lagrangian-Eulerian finite element method, Int. J. Comput. Fluid Dyn., 8(1), p. 1-9, 1997.
[KAO95]
KAO K.-H. and LIOU M.-S., Advance in overset grid schemes: From Chimera to DRAGON grids, AIAA Journal, 33(10), p. 1809-1815, 1995.
[ZHA93]
ZHANG H. et al., Discrete form of the GCL for moving meshes and its implementation in CFD schemes, Computers Fluids, 22(1), p. 9-23, 1993.
[BAT90]
BATINA J.T., Unsteady euler airfoil solutions using unstructured dynamic meshes, AIAA Journal, 28(8), p. 1381-1388, 1990.
[ILI94]
ILINCA A., Calcul des ecoulements compressibles tridimensionnels sur des maillages en mouvement et adaptifs, these de doctorat, Ecole Polytechnique de Montreal, Montreal Quebec, Canada, 1994.
[CRU96]
CRUMPTON P.I. and GILES M.G., Implicit time accurate solutions on unstructured dynamic grids, AIAA-95-1671-CP, p. 285-293, 1995.
[LOE96]
LoHNER R. and YANG C., Improved ALE mesh velocities for moving bodies, Communications in Numerical Methods in Engineering, (12), p. 599-608, 1996.
[FAR94]
FARHAT C. and LANTERI S., Simulation of compressible viscous flows on a variety of MPPs: computational algorithms for unstructured dynamic meshes and performance results, Comput. Methods Appl. Mech. Engrg., 119, p. 35-60, 1994.
[ERI62]
ERINGEN A.C., Nonlinear Theory of Continuous Media, McGraw-Hill Book Company, 1962.
[STO68]
STONE H.L., Iterative solution of implicit approximations of multidimensional partial differential equations, SIAM J. Numer. Analysis, (5), p. 530-558, 1968.
[RAU93]
LEE-RAUSCH E.M. and BATINA J.T., Calculation of AGARD wing 445.6 flutter using Navier-Stokes aerodynamics, AIAA-93-3476, 1993.
A Finite Volume Method for Steady Hyperbolic Equations M. J. Baines, S. J. Leary, M. E. Hubbard Department of Mathematics The University of Reading Reading, RG6 AX, UK
1
Abstract
A finite volume method is presented for steady conservation laws on unstructured meshes which incorporates mesh movement. The method can substantially improve the resolution of sharp features (contacts, shocks) by solving the problem on an optimal mesh. Results are presented for a number of steady state test problems, including scalar advection and the shallow water equations in 2-D. For scalar equations there is a close relationship with the method of characteristics [5]. For shocked flows we describe a discontinuous least squares method which uses a nonlinear shock jump residual to adjust the mesh.
2
Introduction
Finite volume methods for these equations include the cell-based multi-dimensional upwinding schemes. These schemes have been very successful in producing good approximate solutions to the above problems [1] but there is still the possibility of even greater resolution of sharp features by a careful deployment of the mesh [2]. A great deal of effort has been put into mesh refinement near shocks using mesh subdivision, but similar improvements in resolution can also be obtained much more cheaply by minor adjustments to the mesh (see e.g.[3]). In this paper we consider a descent approach using a least squares measure of the residual.
3
Fluctuations
We are interested in systems of steady conservation laws such as the Euler Equations or the homogeneous Shallow Water Equations, of the form div.F(u)=.4(u).Vu=0 where
., .
A(u)
df =^
(1)
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In integral form this becomes
where dS is measured into the arbitrary surface 5. In determining approximate solutions U of these equations we may associate with the differential form (1) the residual R = divf(U) while with the integral form (2) we associate the fluctuation
where ST is a triangle. In fluctuation/distribution schemes such as the Multidimensional Upwind schemes a weighted amount of <j) in each triangle is added to the values of the solution at its vertices. The weights may be chosen so that the schemes are conservative, positive and linearity preserving. In particular, conservation is assured if the weights in each triangle sum to unity. In the Least Squares method, which minimises Mi,
(4)
the use of descent techniques to achieve the optimisation also adds weighted amounts of
4
Counting Issues
On a computational mesh the number of equations given by (2) is equal to the number of cells in the mesh while the number of unknowns depends on the number of nodes in the mesh. In general these are different. When the number of equations exceeds the number of,unknowns, as for example on an unstructured triangular mesh, it is impossible to choose the unknowns to satisfy all the equations. Moreover in any iteration of distribution type in which fluctuations are added to the vertices of the mesh with weights, convergence does not imply that the fluctuations vanish because there is a null space. Likewise, in the least squares approach the norm (4) cannot be driven down to zero because of the existence of the null space. We are interested in allowing the coordinates of the vertices to be additional unknowns of the problem. Typically this will make the number of unknowns exceed the number of equations. In that case, setting all the equations to zero does not determine the unknowns and there are then many solutions which make the fluctuations zero. A fluctuation distribution type of iteration will yield one of the many solutions at convergence.
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A unique solution may be obtained if the number of unknowns is equal to the number of equations. For a scalar equation and a smooth solution this may be achieved on a triangular (or tetrahedral) mesh by including a sufficient number of coordinates per node in the list of unknowns. The fluctuation may then be driven to zero by a fluctuation distribution scheme and the accuracy of the approximate solution depends on the validity of relying on the fluctuation as a measure of the error and the coarseness and/or connectivity of the mesh. We shall discuss the use of Least Squares as a fluctuation distribution scheme in this context. Recall that such a scheme has the advantage of a norm to minimise but is not conservative in the usual sense.
5
A Scalar Problem
We first consider the scalar two-dimensional equation where a is divergence-free. In the least squares method we minimise (4) over both the nodal values Uj and the nodal coordinate Nj in the direction perpendicular to a. We use a steepest descent method of the form
to achieve the optimisation, where Y may be either Uj or Nj and r is a relaxation factor. Fastest convegence occurs when the sweeping takes into account the hyperbolic nature of the original equation. We take scalar examples in which a is a function of x or a function of U. (i) a = (y. —x) in a rectangle — 1 < x < l.Q < y < 1 with initial data
on one inflow side and zero on all others. Results are shown in Figs.l and 2. As expected the norm (4) has been driven down to machine accuracy and the solution shows the characteristics. The redistribution effected by the Least Squares optimisation is attempting to equidistribute
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6
Use of Degenerate Triangles
In the presence of shocks or contact discontinuities least squares methods tend to give oscillatory solutions. A possible elaboration is to introduce degenerate triangles at the shock and to use the Least Squares method with moving nodes to adjust the position of the shock. In effect it is a form of shock fitting. Note that the fluctuation is square-integrable because ^ is always bounded, even at shocks where U is discontinuous. On the other hand, the residual is not square-integrable as a result of divF(U) being unbounded at shocks. That is, on a non-degenerate mesh the l2 norm of the residual is well-defined but on a degenerate mesh it is not. However, the norm (4) is always well-defined. We show results from three examples. (iii) a = (y, —x) in a square — l < x < 0 , 0 < t / < l with initial data
on the inflow side. (iv) a — (u, 1) in a rectangle 0 < £ < l , 0 < y < ^ with initial data
on y = 0. Notice the double value of U at inflow, introducing a contact discontinuity. The degenerate triangles are inserted vertically to connect the triangles on either side of the discontinuity. Results are shown in Figs.3, 4, 5 and 6 (v) The final example is (ii) again in the transcritical case for the flow in a channel, which exhibits a shock in the constriction. An approximate solution is first obtained on a fixed mesh by the method of multidimensional upwinding. This is to approximately locate the shock. Degenerate nodes are then chosen in the approximate position of the shock and the Least Squares minimisation procedure used to position the shock by finding the best position of the shock nodes. Results are shown in Fig.7.
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Numerical results
Figure 1: Initial grid and solution for example(i).
Figure 2: Final grid and solution for example(i).
Figure 3: Initial grid and solution for example(iii).
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Figure 4: Resulting grid and solution for example(iii).
Figure 5: Initial grid and solution for example(iv).
Figure 6: Resulting grid and solution for example(iv).
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Figure 7: Results for example(v)
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8
Conclusion
There exist good methods for the approximate solution of first order PDEs using the fluctuation/distribution technique of Multidimensional Upwinding, achieving conservation, positivity and linearity preservation. However, by the nature of the technique, on unstructured meshes the fluctuations are not driven to zero because of the null space. If we are to rely on the fluctuation to generate a better solution, we need to introduce more degrees of freedom which may be done by adding node locations to the list of unknowns. In this way we can ensure that the fluctuation is reduced to machine accuracy. A convenient way of iterating towards the solution for all variables is to minimise the Least Squares norm of the fluctuation. Although this is not a conservative procedure it is a redistribution procedure and we may think of it as redistributing both the solution and the node locations. The effect is to equidistribute the fluctuation amongst the triangles and so proceed down to the limit in a uniform way. It is useful to regard the procedure as sharpening up a conservative solution. References
[1] H.Deconinck, P.L.Roe and R. Struijs (1993). A Multidimensional Generalisation of Roe's Flux Difference Splitter for the Euler Equations. Computers and Fluids, 22, 215. [2] P.L.Roe (1996). Compounded of Many Simples, in Proceedings of Workshop on Barriers and Challenges in CFD, ICASE, NASA Langley, August 1996, (Ventakrishnan, Salas and Chakravarthy (eds.)), p241, Kluwer, 1998. [3] M.J.Baines. Least Squares and Equidistribution in Multidimensions. Numerical Methods in Partial Differential Equations (1999), to appear. [4] M.J. Baines and S.J.Leary. Fluctuation and Signals for Scalar Hyperbolic Equations on Adjustable Meshes. IJNME (submitted). [5] M.J.Baines and M.E Hubbard. Multidimensional Upwinding with Grid Adaptation. In Numerical Methods for Wave Propagation (E.F.Toro and J.F.Clarke (eds.)), Kluwer (1998).
Moving grid technology for finite volume methods in gas dynamics Boris N. Azarenok and Sergey A. Ivanenko Computing Center of RAS, Vavilov str. 40, GSP-1, Moscow, 117967, Russia e-mail: [email protected] , [email protected] ABSTRACT Adaptive grid generation procedure is coupled with the Godunov-type solver. The problem to construct the harmonic coordinates on the surface of the graph of control function is formulated. The projection of these coordinates onto a physical region produces an adaptive-harmonic grid. Results of computations for the nonstationary flow in the wind tunnel containing a step are presented. Keywords: Adaptive grid, harmonic mapping, numerical methods in gas dynamics
Introduction Recent development of robust adaptive grid generation techniques which can give locally refined information in areas of interest as well as fitting boundaries precisely is of great importance for the practical problems, particularly for simulating the moving shocks in the nonstationary problems of gas dynamics. In the present study the r-refinement adaptation is used with the Godunovtype finite-volume solver of the second order accuracy to model a gas flow. The approach is based on the grid cells convexity concept [2] and the theory of harmonic maps between surfaces. The harmonic functional (Dirichlet's functional) is approximated in such a way that its minimum ensures all grid cells to be convex quadrilaterals. The main property of this approximation is that the discrete functional has an infinite barrier on the boundary of the set of grids with all convex cells and it guarantees nondegenerate (unfolded) grid generation at each time step. This folding-resistant property is very important in the problems with interior lines following thin and strongly bent layers of high gradients. Another important property is that the grid generator is solver-independent. 1. Governing Equations and Finite-Difference scheme We use the governing conservation laws expressed in integral form to describe the planar flow of the ideal gas. The volume integrals in the space x—y—t may be transformed to the surface integrals by virtue of Gauss's theorem as it is shown below for the law of conservation of mass
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Thus, a variational formulation of the problem is written as following [41 :
where the vectors of conserved variables U, F(U) and G(U) have the following components
here u and v are the velocity components, p and p are the pressure and density. Let us introduce the moving grid in space x—y—t. Integrating the system over the oriented surface being the boundary of the computing cell we get a cell-centered finite-volume discretization of the governing equations
where of the cell at /
are n+1
n
and t .
the areas of the upper and lower faces
Each of four vector values
ls an avera
' ge flux of mass, impulse and energy through the corresponding intercell surface towards the outside normal vector. To calculate the values at the next time level n+1 we use two-step procedure "predictorcorrector" applying a monotonicity algorithm and solution of the Riemann problem [1]. The scheme is of the second-order accuracy in time and space. 2. Dynamic solution-adaptive mesh algorithm Suppose we are given a simply connected domain Q with a smooth boundary in the x,y plane. Consider the surface z = f ( £ , y ) of the graph of the control function f ( x . y ) . It is required to find a mapping of the parametric square onto the domain Q under a given mapping between boundaries such that the mapping of the surface onto the parametric square be harmonic. Thus, the problem is to minimize the harmonic functional which can be written in the form
Adaptivity, tracking and fitting 797 This functional defines an adaptive-harmonic grid clustered in the regions of high gradients of the function f ( x , y). Detailed account of the minimization algorithm is presented in [2]. One time step to solve the 2-D equations of gas dynamics with automatic grid adaptation contains the following stages: 1. Generate the mesh at the next time step. As an initial approach we can either use the mesh from the previous time level or, that is most effective, define the node coordinates by their velocity from the previous time level. 2. Compute the gas dynamics values at the next time level using the secondorder Godunov-type scheme described in sec.l. 3. Update the gas dynamics value chosen as a control function from the cell centers to nodes. The result is a control function value /,-j at each mesh node. 4. Evaluate the derivatives (fx)ij and (fy)ij at each mesh node. 5. Make one iteration step and compute the new values of Xij and yij. 6. Repeat starting with step 2 to convergence. Note that this approach can be extended to the general structured multiblock grid configurations. 3. Results of test computations As a test example the planar nonstationary flow in the wind tunnel containing a step [3] has been considered. The wind tunnel is 1 unit wide and 3 units length, step is 0.2 units height and begins at x=0.6. Initial conditions for ( u , v , p , p ) are ( 3 , 0 , 1 , 7 ) that corresponds to the flow-in boundary condition at x—0 as well. Here 7 is the ratio of specific heats and equal 1.4. The exit boundary has transmissive conditions since at x=3 the flow is supersonic. Along the walls of the tunnel reflecting boundary conditions are applied. The corner of the step is the center of a rarefaction fan and hence is a singular point of the flow. Just above the step there is a thin boundary layer. Shock wave interacts with this layer and qualitative nature of the flow near the step is altered. When computing on the rectangular mesh we apply the additional boundary condition to the density and velocity near the corner of the step [3] to minimize numerical errors generated by the corner. Results of computations on the uniform and rectangular mesh with steps hx—hy = 1/160 are presented in Fig.l. The mesh contains 480x160 cells. Isolines of the density are shown in Figs.la-d corresponding to time moments t—0.5, 1, 2,4. The results of Fig.l are the same that in [3]. At time 2=0.5 (Fig.la) the bow shock is formed, then extending by t=l it reaches the top wall and reflects from it (Fig.lb). In some time, at t=2 (Fig.lc), the shock reflects from the step and near the top wall the Mach stem emerges and the contact discontinuity emanates from the triple point. By time t=4 (Fig.Id) the shock wave has already reflected triply, the contact discontinuity twice intersects the shock. In Fig.2 the results of computations on the adaptive moving grid 150x60 with applying the procedure to capture the bow shock are presented at t=4.
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Fig.2a,b corresponds to adaptation with the density as the control function. In case, presented in Fig.2c,d, we perform adaptation to \V\. The density contours are shown in both cases. Note that the Mach stem is described rather precisely by the moving boundary but the shock, emanating from the triple point, is not resolved well by nodes clustering. In the first case of adaptation to the density (Fig.2a,b) the procedure does not provide grid clustering to the contact discontinuity. At the same time adaptation to \Vr\ produces weak grid clustering to the contact discontinuity (Fig.2c,d). In the case of global adaptation without shock capturing the method produces excellent results shown in Fig.3,4 at the same time moments as in Fig.l. The mesh contains 180x60 cells that seven times less than for the rectangular grid (Fig.l) and here at the same accuracy it is required less computer time by factor of 1.5. Here, like for the uniform grid, the shock wave is smeared within 3 or 4 cells and the mesh with strong clustering looks like a set of the blocks with the boundaries defined automatically as the lines of grid nodes clustering. Inside every block the mesh is quasiuniform and the solution is smooth, see Fig.4c. This approach may be considered as further development of the moving block technology presented in the monograph [4]. Advantage of the moving adaptive grid technology is "the automatic generation of moving block boundaries" that can not be done by hand in many cases. This problem is essential difficult in three-dimensional cases. 4. Conclusions Modeling of the nonstationary flow in the wind tunnel containing a step has shown that the accuracy of the solution obtained on the adaptive grid increases greatly in comparison with the rectangular grid. Numerical experiments confirm the high reliability and accuracy of the algorithms which employ adaptive-harmonic method to construct the grid. It is shown that for the problems with moving shocks adaptive moving grids appear to be more economical, at the same resolution as on the rectangular grid, and in our case reduce the memory requirements by the factor of 10. Presented solver-independent solution-dependent grid generation technique is stable and robust. In its application we can avoid human intervention into the solution process. It can be extended to the three-dimensional case and can be used in other problems of computational fluid dynamics. References [1] Azarenok B.N. On a realization of high-order Godunov's scheme, Computing Center of RAS, (Moscow, 1997). [2] Ivanenko S.A. Harmonic mappings, Chapt. 8 in: Handbook of Grid Generation. J.F. Thompson at al, (CRC Press, Boca Raton, Fl, 1999). [3] Colella P. and Woodward P.R. J.Comput.Phys., 54 (1984), pp. 115-173. [4] Godunov S.K. at al, Numerical solution of multi-dimensional problems in gas dynamics (Nauka Press, Moscow, 1976).
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Fig. 4
Numerical Simulation of Lifted Turbulent MethaneAir Diffusion Flames
Ming Chen, Nobert Peters Institut fur Technische Mechanik, RWTH-Aachen, Templergraben 64, D-52056 Aachen, Germany
ABSTRACT This paper presents the numerical simulations of the lifted turbulent methane-air diffusion flames using a flamelet concept. To describe turbulent flame propagation and stabilization in inhomogeneous mixtures, combined flamelet formulations for partially premixed flames have been applied, which include transport equations of the mean mixture fraction Z and its variance Z" , and an equation for a smooth scalar field G, determining the mean location of the flame front. The turbulent burning velocity for partially premixed combustion is described. The level set approach is applied to solve the G-equation. The simulations suggest that the mechanisms of stabilization can be explained by the partially premixed flame propagation and flamelet quenching. The stabilization point is determined by the local flow velocity, the local turbulent burning velocity, and the local scalar dissipation rate. The stabilization points are found not to be located at the isoline of the stoichiometric mixture, but at the lean side. Key Words: flame lift-off
, turbulent combustion, numerical simulation.
1. Introduction Despite a lot of research efforts, the physical mechanisms of the flame stability are still not clearly explained. Pitts [PIT 88] has reviewed the theories and experimental findings of the lifted diffusion flames. The mechanisms responsible for the flame stability may be divided into three groups: 1) premixedness model; 2) flamelet quenching model; and 3) flame extinction due to large and small scale turbulent structures. The premixedness model [VAN 66] assumed that fuel and air along the
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stoichiometric mixture are fully premixed and flame stabilization occurs at the position where the axial mean velocity equals the turbulent mean burning velocity. However, the premixedness model has been challenged by several new theories [PET 83, BYG 85, MUL 94], which argued that the stabilization of the lifted diffusion flames can be explained by flame extinction processes. Peters and Williams [PET 83] proposed that non-premixed combustion occurs and flame propagation proceeds along instantaneous surfaces of stoichiometric mixtures up to the point where the flame is quenched. Byggsoyl and Magnussen [BYG 85] proposed that the extinction occurs in the smallest vortices of the turbulent flow. Broadwell et al. [BRO 85] suggested a large scale mixing model to explain the lift-off and blowout behaviors. A numerical calculation of lifted diffusion flames has been done by Miiller et al. [MUL 94], they used the combined flamelet formulations for premixed and non-premixed combustion to calculate flame propagation and lift-off heights in jet diffusion flames. In this work, the numerical calculation of the lifted turbulent methane-air diffusion flames have been carried out with the purpose to gain more understanding of the lift-off behavior from the point of view of numerical simulation. The measurements by Tacke [TAG 98] showed that the stabilization point of a lifted methane diffusion flame is not located at the isoline of the stoichiometric mixture, but at the lean side, which is contrary to the expression of the turbulent burning velocity used by Miiller et al.. To investigate the position of the stabilization point, the formulation of the turbulent burning velocity applied in the present study is more general. Following a describtion of the combined flamelet formulations of partially premixed combustion, numerical methods will be introduced. Finally the results of the calculation are presented and discussed. 2. Mathematical Model The flamelet model [PET 87] has been proven to be very useful to describe non-equilibrium chemistry in combustion systems in the flamelet regime, when chemistry is fast compared to the mixing time scales of the flow. The concept allows to decouple the effort to describe combustion chemistry from the calculation of the turbulent flow field. Originally, flamelet models have been developed to treat non-premixed combustion problem [PET 87], and later premixed flames [WIR 92]. In order to describe turbulent flame propagation and stabilization in inhomogeneous mixtures, both approaches have been combined by solving scalar equations of the mean mixture fraction Z, its variance Z"2, and a mean G-field equation [MUL 94].
2.1. The flamelet model for non-premixed combustion Additionally to the basic conservation equations for mass, momentum, tur-
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bulent kinetic energy k and the dissipation rate of turbulent kinetic energy £, the transport equations of the mean mixture fraction and its variance should be solved, which read
where the Schmidt numbers Sc^ and Sc—^ are constants. The scalar dissipation rate \ ls modeled as
The energy conservation equation is solved in the form of the total enthalpy:
Here PTT is the turbulent Prandtl number, and q'n denotes a radiation heat loss term. 2.2. The G-Equation for partially premixed combustion To describe the location of the pemixed flame front, the scalar function G is applied in the flow field. The scalar G is so defined that G = G0 describes the location and geometry of the flame front. A mean turbulent G-equation may be derived from the kinematic consideration [PET 97]:
where the mean flow velocity vu is defined as the conditional velocity in the unburnt mixture ahead of the flame front and vu' is its fluctuation. 5° is the laminar burning velocity of a unstretched planar flame, K is the flame curvature and DC = S°LL is the Markstein diffusivity. The absolute value of the gradient of the scalar G is represented by a = |VG|, and the normal vector n is defined as n = — VG/|VG . It has been shown that Equation (5) can be modeled as [PET 971
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where the diffusivity DT-Q is H T / S C T , G . The term S0L can be expressed as
where P(Z, x-, &} is a joint pdf of Z, x and cr. It is assumed here that Z, x and a are statistically independent, and the fluctuation of the scalar dissipation rate can be neglected, then one obtains
Letting |VG| = /0°° P(cr)cr da and using an ad-hoc expression that modifies the laminar buring velocity 5£(Z) by a quenching term, in terms of the scalar dissipation rate x [MUL 94]
one obtains,
The scalar dissipation rate at quenching Xq is 6.4 s-1 for methane-air diffusion flamelet. With the relation following Wirth M. et al. [WIR 921
Equation (10) can be written as
Here soT denotes the turbulent burning velocity for premixed flame. With the pYnrpssinn [WTR 92]
one obtains
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Here the constants 61 = 1.5, b2 = 0.8,63 = 1.4 are obtained from experiments and direct simulations. Instead of using Equation (3), the scalar dissipation rate x inEquationn(14) is modeled following [101:]:
where AZ/? is the flame width in mixture fraction space, which is assumed as AZ/? = 2Zst. 1.3. Treatment of the partially premixed combustion The present method treats the discontinuous variation of all states as coninuous crossing from unburnt to burnt states in a region sealed with turbulent lame brush. The weighted value of the mass fraction of species i is calculated by The fraction of burnt flame in each cell /& is obtained by assuming that G fluctuations satisfy a Gaussian distribution:
where the variance of G is assumed proportional to the squre of the integral length scale with a proportionality factor of 1/4. The local mass fraction of unburnt mixture Fj jU is obtained by a quenched diffusion flamelet while the mass fraction of burnt mixture Y^ is obtained from the burnt diffusion flamelet. The mean mass fraction of species of burnt state is obtained by weighting the laminar values with an assumed udf, i.e. a beta-ndf.
The code structure for partially premixed combustion is shown in Fig. 1. 3. Numerical methods for solving G-equation
In this work, the level set approach of Sussman et al. [SUS 94] is used to solve the G-equation. Instead of explicitly tracking the interface, the level set approach captures the interface implicitly, in which the interface is identified as the zero level set of a smooth function. To guarantee that G is a smooth function in space, it is useful to keep G a distance function. Therefore, when G = GO, the condition |VG| = 1 has to be satisfied. The smoothing algorithm of Sussman et al. is used solving the following equation to a steady state:
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The application of a conservative grid adaptation technique to ID unsteady problems
Manuel J. Castro-Diaz Numerical Analysis. University of Malaga 29080 Malaga Pilar Garcia-Navarro Fluids Mechanics University of Zaragoza
ABSTRACT: In this contribution, we present an efficient conservative mesh adaptation algorithm applied to 1-D shallow water equations. This algorithm is suitable for unsteady situations and discontinuities of the solutions are well captured. Numerical results are presented. Key Words: Grid adaptivity, Q-Schemes of Roe and Van Leer, Implicit and explicit TVD methods, unsteady shallow water flows.
1. Introduction Numerical methods for predicting the water profile and discharge in steady as well as unsteady situations of hydraulic systems have become a common tool. In particular, finite difference applications of numerical schemes have been widely reported. One of the basic problems in unsteady hydraulic systems is the location of solution discontinuities and shocks. In order to solve this problem, an efficient conservative grid adaptation algorithm applied to the resolution of the shallow water equations is presented. First, the equations to be solved are presented. They are essentially the well known shallow water equations written in conservative form. The discretization of the system is done using the numerical method proposed by BermudezVazquez in [BER 94]. A high order method as the TVD-McCormack scheme (see [GAR 92-1] and [GAR 92-2]) has also been used to compare numerical solutions.
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A posteriori error estimator to control the error of the numerical solution is constructed using a metric tensor M, being M the solution of a minimization problem. Once M is computed, a simple version of an anisotropic Delaunay algorithm for one-dimensional domains is used to adapt the mesh. A local conservative interpolation algorithm is used to guarantee conservation of the variables during mesh adaptation. Finally numerical results are presented and comparisons with non conservative interpolation algorithms and other numerical schemes are given. 2. Shallow water equations Shallow water equations represent mass and momentum conservation along the direction of the main flow. They constitute an adequate description for most of the problems associated with open channel flow modelling and can be written as the following system of equations:
where A is the wetted cross sectional area, Q is the discharge, g is the acceleration due to gravity and I1 represents an hydrostatic pressure force term:
and I2 accounts for the pressure forces due to longitudinal width variations,
beeing B ( x ) the breadth of the channel that it is supposed to be locally rectangular. The right hand side of equation (1) also contains the sources and sinks of momentum arising from the bed slope and friction losses. 3. Discretization of the equations The discretization of the system is done using the numerical method proposed by Bermudez-Vazquez in [BER 94], that is, an explicit extension of the
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Q-scheme of Roe with upwinding in flux and source terms. For more details see [ROE 81], [ROE 86] and [BER 94]. An high order method as the TVDMcCormack scheme (see [GAR 92-1] and [GAR 92-2]) has also been used to compare numerical solutions.
4. Error estimator: metric computation A posteriori error estimator to control the error of the numerical solution is constructed using a metric tensor M, being M the solution of the following minimization problem (see [CAS 97] and [CAS 96] for more details): Find a metric tensor M, so that, the adapted Delaunay mesh constructed from M minimizes the interpolation error
where, Wn = (An,Qn}T is the solution of the shallow water equation at time t — tn and n/Jpy™] is a continuous piecewise linear interpolation of Wn over the mesh TV In general, this metric tensor is given in terms of Hessian matrices of the variables. In this particular case, as An and Q n :IR —> IR, the metric tensor can be computed as:
where D2An and D2Qn are, respectively, the second derivatives of An and Qn, e0 is a positive control parameter and T^1 a truncated function that avoids metric degeneration
Where, USUally, C0 = l/l^ax
and
e
l = Vmtn' bem§ lmaxx and lmin, the maximalll
and minimal allowed length for mesh edges, respectively. Note that An and Qn are unknowns as they are the solution of the problem at time t = tn, there fore, the metric tensor M is approximated using the numerical solution at time t = tn, A%, Q%. For more details and the extension to bidimensional domains see [CAS 97] and [CAS 96].
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5. Conservative mesh adaptation algorithm
Once the metric tensor M. is computed, the mesh is adapted using a anisotropic Delaunay algorithm. For one-dimensional meshes the algorithm is simple (see [CAS 97] and [CAS 96]): let di the length of the edge a^ with respect to the metric tensor M.. Three possible cases can be distinguished: • If di > dmax (dmax w 1.4) then a^ is cut in two edges. The length of the new edges is computed and they are split until the length of all the new edges is smaller than dmax. • If di < dmin (dmin « 0.6) then the edge a^ is suppressed. As this process implies that neighbour edges have changed, we must check if their lengths are larger than dmax. In that case, the previous step is applied to the corresponding edges. • If dmin < di < dmax, &i IS kept.
One of the most difficult problems on mesh adaptation is the interpolation of the numerical solution onto the adapted mesh. This is critical if the studied phenomena are unsteady. A deficient interpolation could spoil the good properties of the numerical scheme, as conservation and monotonicity. The usual interpolation operator in mesh adaptation is the linear one, but this operator, in general, is not conservative, that is, given the numerical solution of the shallow water equations over the mesh Th at time step n,(A£, Q£),
being H/^ [A£], n/^QJJ] a continuous piecewise linear interpolation of (A£, Q£) over the adapted mesh, T^, at time t = tn. In order to guarantee the conservation of variables during the mesh adaptation process we propose the following interpolation operator: Tlch, [A%] is the continuous piecewise linear function over the mesh TV such as
where, v'j is a vertex of TV , C'j is the cell associated to v'- and \C'j \ is the length of cell q. Note that fc, nT A% could be difficult and expensive to calculate if Th ^ T^. We can avoid this problem if we perform this computation during the mesh adaptation loop since C'j fl T^ is easy to determine. Using conservative interpolation together with mesh adaptation, the discretization error and CPU time can be reduced substantially for unsteady problems (see table 1).
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6. Numerical results 6.1. Dam break problem This is an interesting problem to test the efficiency of conservative mesh adaptation algorithm for non-steady flows with shocks since it has an analytical solution. This problem is generated by the homogeneous one-dimensional shallow water equations with the initial conditions
In this case, hL = 2, HR — 1 and L = 60. If the calculation times used are so as to avoid interaction with the extremities of the channel, the boundary conditions are trivial. Table 1 summarizes the results obtained for the Q-scheme of Roe with an uniform mesh, non-conservative adaptation, conservative adaptation and for the TVD-McCormack scheme (TVD-MC) with uniform mesh at time t = 2.5 seg. As can be observed, the Q-scheme of Roe+conservative adaptation only needs 332 nodes to obtain an error of 0.053 units. If an uniform mesh is used, the number of nodes must be about 3000 for a similar error. For a higher order scheme as the TVD-MC the number of nodes is about 1000. Note that, if a non-conservative mesh adaptation algorithm is used, the approximation error increases up to 0.0583 units. The reduction of CPU time for a similar tolerance error is significant, if mesh adaptation is used and note that the computational cost for conservative and non-conservative adaptation is practically the same. Figure 1 shows a comparison between the numerical solution for the dam break problem with conservative mesh adaptation and the exact solution at time t = 2.5 seg. Adaptation No No conservative non-conservative
scheme Roe TVD-MC Roe Roe
N. Nodes 3000 1000 332 325
L 2 -Error 0.053487 0.052360 0.053869 0.058308
CPU Time 164.45 seg. 23.3 seg. 16.16 seg. 16.20 seg.
Table 1: Table 1: Dam break problem. 6.2. Surge propagation through converging-diverging channel In this example, the geometrical domain for the flow is an interval of L — 500 m with flat bed and a sinusoidal width variation given bv 'z-25(V :-2E B(x) = i 5 - 0.7065(1 +cos (27T (^2))} if \x - 250| < 150 300 ^ 5 otherwise
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Figure 1: Dam break problem with conservative mesh adaptation. Comparison with the exact solution at time t=2.5 seg. /IL = 2, HR = 1. In this case the exact solution does not exist and only comparisons with other schemes can be performed. The time evolution of a surge is considered. A bore 9.79 m deep of 1000 cum/s propagates downstream over still water 1 m deep. A 2 m weir is supposed to be placed downstream. At time t = 150 s the downstream end is reached by a front similar to the initial one so that it is partially reflected and partially transmitted over the weir (see figure 2). Only 142 nodes are needed to obtain a good approximation if conservative mesh adaptation is used, while a uniform mesh of 600 nodes is needed if TVD-McCormack is used. The reflected surge starts travelling upstream and it propagates until it becomes a stationary hydraulic jump in the contracting region. This final steady state is shown in figure 3. The total CPU cost when using mesh adaptation is 186 seconds while the total CPU time when using TVD-McCormack is 360 seconds in a PENTIUM II (333 Mhz.) 7. Conclusions
Conservative mesh adaptation has been applied with success to 1D unsteady problems. The idea of the method is simple and it can be easily applied by other users. The application to 2D and 3D configuration is straightforward. Conservative interpolation guarantees the conservation of all the variables during mesh adaptation and, as the results show, the numerical error is reduced when it is used. The global CPU requirement is significantly reduced compared with a direct computation on a uniform fine mesh. Acknowledgements: The author is indebted to J. Macias and E. Vazquez
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Figure 2: Surge propagation in a converging-diverging channel. Comparison between conservative adaptation and TVD-MC scheme at time t=150 s.
Figure 3: Surge propagation in a converging-diverging channel. Comparison between conservative adaptation and TVD-MC scheme at time t=600 s.
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for many valuable discussions. This work has been supported by the Comision Interministerial de Ciencia y Tecnologia (C.I.C.Y.T.), project MAR97-1055CO2-01. 8. Bibliography [BER 94]
BERMUDEZ, A., VAZQUEZ, M.E. Upwind methods for hyperbolic conservation laws with source terms. Computers and Fhmfc,23(8),1994, p. 1049-1071.
[CAS 97]
CASTRO. M.J, F. HECHT, B. MOHAMMADI AND O. PIRONNEAU. Anisotropic Unstructured Mesh Adaptation for Flow Simulations. I.J. for Numer. Meth. in Fluids, 1997, 25, p. 475-491.
[CAS 96]
CASTRO-DlAZ, M. Generation y Adaptation Anisotropa de Mallados de Elementos Finitos para la Resolution Numerica de E.D.P. Aplicaciones. Ph.D. Universidad de Malaga. 1996.
[GAR 92-1]
GARCIA-NAVARRO, P. AND ALCRUDO, F. Implicit and Explicit TVD Methods for Discontinuous Open Channel Flows. Proc. of the 2nd Int. Conf. on Hydraulic and Environmental Modelling of Coastal, Estuarine and River Waters, Vol. 2, Edited by R.A. Falconer, K. Shiono and R.G.S. Matthew, 1992, Ashgate.
[GAR 92-2]
GARCIA-NAVARRO, P., AND ALCRUDO, F. ID Open Channel Flow Simulation Using TVD McCormack Scheme. J. of Hydraulic Engineering, ASCE, 118, 1992, p. 1359-1373.
[MCC 71]
McCORMACK, R.W. Numerical Solution of the Interaction of a Shock Wave with a Laminar Boundary Layer. Proceedings of the 2nd International Conference on Numerical Methods in Fluid Dynamics, 1971, p. 151-163.
[ROE 81]
ROE, P.L. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics, 43, 1981, p. 357-371.
[ROE 86]
ROE, P.L.Upwinding differenced schemes for hyperbolic conservation laws with source terms. In Carasso, Raviart, and Serre, editors, Proceedings of the Conference on Hyperbolic problems, Springer, 1986, p. 41-51.
Application of mesh adaptive techniques to mesh convergence in complex CFD
David Leservoisier(*)(**), Alain Dervieux(**), Paul-Louis George(***), Olivier Penanhoat(*)
(*) SNECMA, Centre de Villaroche, 77550 Moissy Cramayel, FRANCE (**) INRIA Sophia-Antipolis BP 92, 06902 Sophia Antipolis Cedex, FRANCE (***) INRIA Rocquencourt BP 105,78153 Le Chenay Cedex, FRANCE
ABSTRACT This paper is devoted to the illustration of the fact that mesh adaptation is a natural path for mesh convergence for complex problems. We apply an anisotropic reconstruction mesh adaptor. The field of application considered is that of turbulent compressible flows.
1 Introduction It is a remarkable fact that up to now nearly only second-order methods have been efficiently and routinely applied to complex CFD such as RANS (ReynoldsAveraged Navier-Stokes) finite volume simulations of compressible flows. Secondorder accurate schemes are evidently much better than first-order accurate ones, but pioneering experiments in applying higher order schemes to practical configurations are often disappointing. In fact, the actual order of convergence of both - second and higher order- finite volume methods on practical problems is questionable; indeed, transonic flows involve singularities for which numerical schemes produce only first-order accurate results. Even nonlinear limited schemes (TVD, etc.) have permitted second-order accuracy on most part of the domain but they do not capture a shock with an approximation error better than first-order in the usual L2 (root mean square) functional norm. Then, making the spatial mesh finer will make the error on the discontinuity grow bigger that the rest of the error and decide the size of the asymptotic error of the global calculation, that is, 0(Ax). These discontinuities are in fact the prototype of any detail (smooth or not) of the flow that is not accurately represented by the mesh. Think of wakes, slip lines or surfaces, boundary layers,... In most complex CFD calculations, the rriesh is not fine enough for capturing well these details; moreover, numerical dissipation makes this situation worse; as a result, second-order convergence to continuous solutions is generally not observed. The purpose of this paper is to
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study some conditions that would allow actual second-order convergence for 2D RANS CFD calculations. The key point on which we shall rely is the use of mesh adaptation. It is clear that for a lot of complex calculations in which small details arise in very localised areas of the computational domain, the rnesh is in fact one of the main unknowns of the numerical problem. This explains why in many new adaptive approaches, the mesh is globally reconstructed instead of only modified in order to be better adapted to the solution. In [3], the authors describe a method in which a local matrix called "metric" is computed from an estimate of the hessian derivative of the unknown; then a new anisotropic mesh is rebuilt by a Delaunay technique taking account of the adaptation metric. In this study, we propose, as a complement of this method, a strategy for researching mesh convergence. For a given number of unknown (degrees of freedom) we make correspond a "discrete answer" defined as the association of a adapted mesh and a corresponding EDP discrete solution. We suggest that this discrete answer allows second order mesh convergence in many cases where the usual mesh refinement produce only first-order convergence. Further we claim that this strategy allows mesh convergence for complex problems such as industrial CFD flows. This assertion will be illustrated by several turbulent flow calculations.
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Mesh adapted solution
In the recent bibliography on mesh adaptation, it is recognized that mesh adaptation is a mean for compensating lack of accuracy due to singularities and steep gradients in the solutions. Quantitative assessment of this property is possible by considering the number of nodes instead of mesh size: for singularities, second-accuracy is observed, while for regular gradient stiffness, second-order accurate behavior arises with a considerably much smaller number of nodes than for uniform meshes. For a rather simple presentation of these facts, see [1], in which the distribution of a given number of nodes in an interval. Let us define an adaptive algorithm which associates to a given number of nodes N the couple (A^N^N) of a adapted mesh and a numerical solution. The solution UN is computed on the mesh MN, and the mesh MN is adapted to the solution UNMesh adaptation: The principle of mesh adaptation is an already rather popular combination of Hessian evalutation and anisotropic mesh reconstruction with total mesh size control [3].
2.1
Application to adaptive interpolation
The second-order convergence to continuous solution depends of both the adaptive process and the approximation of PDE.
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Mesh coupling: In order to obtain the above described couple, we parametrise
Figure 1: Adaptation to an analytic "eight-shaped" function: isotropic and anisotropic algorithm. the remesher in order to build a new adapted mesh with (in fact approximatively) the prescribed number of nodes. Then, a new solution of the PDE is computed and produces a new metric. The process is repeated until convergence ; a sufficient iterative convergence can be recognised if we observe a rather small, but not so small, variation of the PDE solution between two meshes. The rest
Figure 2: Mesh convergence for the interpolation of the eight shape function on a sequence of uniform meshes: multiplying the number of nodes by four does not produce a solution four times more accurate. of the paper will concentrate on the verification that the proposed algorithm allows second-order accurate convergence towards the continuous solution, that is, for a norm of integrable functions (typically : L 2 ), \\Ufj — U\\ = O(N°'5}.
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For a first study that be independant of the approximation of the PDE, we consider first replacing the solution of the PDE by a simple P1 interpolation of an analytical function. Examples of isotropic adapted mesh and anisotropic adapted mesh are depicted in Figure 2. The given fonction has steep gradients along an eight-shaped curve. For each element of a sample of values of N, number of nodes, we have iteratively converged the sequence described in Fig.l (in which "solution of EDP" is replaced by "interpolation of analytic function"). This process yields for each TV a couple ( M N , U N ) and an error CN = \\UN — ^ I k 2 - Convergence curves of this error to zero for the uniform, isotropic adaptive arid anisotropic adaptive strategies are depicted in Figures 2, 3 and 4. Second-order convergence
Figure 3: Mesh convergence for the interpolation of eight-shaped analytic function with a sequence of adapted isotropic meshes (both coarse and finer meshes are perfectly adapted by iterating mesh-solution coupling): second-order accuracy appears only if more than 10000 nodes are used. to continuous solution appears already for a small number of nodes when the anisotropic adapted algorithm is applied, see Fig. 5. With the isotropic algorithm, second-order convergence appears only for number of nodes larger than 5,000 (Fig. 4). Second-order convergence with the uniform algorithm is not obtained, even with 30,000 nodes Fig. 3.
3
Application to turbulent flows
We now concentrate on assessing the described strategy on a set of typical flows of industrial interest. These flows will be computed with a 2D compressible flow solver with k — 6 turbulence modelling and wall law, relying on a MUSCL approximation on unstructured meshes with triangles, see [2] for details. The hessian sensor is computed with the six conservative variables, including
Adaptivity, tracking and fitting
821
Figure 4: Mesh convergence for the interpolation of the eight-shaped analytic function with a sequence at adapted anisotropic meshes: Assuming you have at least 100 nodes, four times more nodes brings four time more accuracy. density, moments, energy, turbulent kinetic energy, turbulent dissipation rate.
3.1 Flat plate flow We start with the capture of -a part of- a boundary layer by computing a supersonic flow past a flat plate (with the wall law, only the fully turbulent part of the boundary layer is captured). We consider the friction velocity on a point after the stagnation point. Two strategies are compared: the isotropic one arid the anisotropic one. We observe that the isotropic strategy indeed allows second-order convergence, but only for meshes finer than 15,000 nodes. The anisotropic strategy give second-order convergence already for 6,000 nodes.
3.2
Back-step flow
An interesting prototype of such flow is the famous back-step flow [4]. We concentrate on an important but difficult output, the abscissa of the reattachment point. Results are sketched in Fig. 7. We observe that the results are no so good since the convergence order is only slightly more than one. In fact such an output as the position of a vanishing point is rarely approximated with the accracy of other more standard output.
4
Concluding remarks
This paper discusses the research of mesh convergence for a class of industrialtype flow calculations. We claim that mesh adaptation is the key for secondorder convergence (or further) for such calculations. This is illustrated by a
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Finite volumes for complex applications
Figure 5: Mesh convergence of friction velocity for the flat plate case with isotropic adaptation
Figure 6: Mesh convergence of friction velocity for the flat plate case with anisotropic adaptation (dots for second-order, dashes for first order slopes)
Adaptivity, tracking and fitting
823
Figure 7: Mesh convergence reattachment abscissa for the back step flow with anisotropic adaptation few examples. For the plate flow, we illustrate that already fine meshes are necessary for observing second-order convergence. With anisotropic adaptation, the necessary fineness is less (by a factor two) than with the isotropic one. For the prediction of a reattachment point, even with mesh adaptation, second-order convergence is not obtained. We emphasize that the model choosen is a non stiff one (from the approximation point of view) in accordance to the common opinion, and also as shown Carrau and Dervieux in [7]. In despite of this, the size of necessary meshes for secondorder convergence is large. We can expect an amplification of this property for low-Reynolds modeling and of course 3D calculation. In both cases, the application of anisotropic mesh adaptation seems to us mandatory.
References [1] Palmerio, B. and Dervieux, A. Multirnesh and multiresolution analysis for mesh adaptive interpolation. Applied Numerical Mathematics, 22 (1996). [2] Carre, G. An implicit multigrid method by agglomeration applied to turbulent flows. Computers and fluids, 26, No. 3, pp 229-320, (1997). [3] Castro-Diaz, M.J., Borouchaki, H., George, P.L., Hecht, F. and Mohammadi, B. Anisotropic adaptative mesh generation in two dimensions for CFD. Computational Fluids Dynamics '96, pp 181-192, (1996). [4] Look, P.H., McDonald, M.A. and Firmin, M.C.P. Pressure distributons and boundary layer and wake Measurements. AGARD AR 138, A6-1 A6-77, (1979).
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Finite volumes for complex applications
[5] Laug,P., Borouchaki, H. and George, P.-L. Maillage de courbes gouverne par urie carte de metriques. INRIA Rocquencourt, Rapport de recherche n° 2818 (1996). [6] Dhatt, G. and Touzot, G. Une presentation de la methode des elements finis. Collection Uriiversite de Compiegne. Maloine S.A. Editeur [7] Carrau, A. and Dervieux, A. Application of a mesh adaptive capture strategy for solving ID turbulent layers", Proc. of the Fourth Internatioonal Conference on Numerical Grid Generation in Computational Fluid Dynamics and related topics, Swansea, UK, 6-8 april., 1994, N.-P. Weatherhill, P.-R. Eiseman, .1. Hauser, J.-F. Thompson (eds.), Pineridge Press Ltd., 513-526, Swansea, Wales (1994)
Figure 8: Obtaining the adapted couple (mesh, solution): Better adapted rneshes, with (about) the same number of nodes and new flow solution on these meshes are iteratively rebuilt until mesh and solution do not vary much.
Multidimensional Fully Adaptive Finite Volume Schemes for the Numerical Simulation of Stiff Combustion Front Propagation in Condensed Phase
A. Aoufi LIMHP-CNRSS Avenue JB Clement 93430 Villetaneuse France
ABSTRACT A mathematical modelling of multidimensional solid-solid combustion propagation based on a nonlinear reaction-diffusion equation is presented. A fully adaptive, finite-volume scheme on a structured non uniform mesh is detailed. The computational efficiency of this numerical scheme is discussed in the case of several multidimensional simulations. Key Words: Reaction-Diffusion Equation, Adaptive Finite- Volume, Adaptive Time-Stepping, Phase-Change Modelling, Solid-Solid Combustion.
1. Introduction
We consider the mathematical modelling of the ignition and the propagation of stiff combustion fronts in SHS ( Self-propagating High Temperature Synthesis) process of Ti + C —>• TiC in various geometries - slab, cylinder and parallelepipedic volume-, by a reaction-diffusion equation coupled with a stiff differential equation for the kinetics . This solid-solid reaction propagates inside a sample (made of a stoechiometric mixture of micronic titanium and carbon particles) if sufficiently enough heat is supplied on a significant part of its exterior surface. We will focuss in this paper on the description of the numerical scheme we have designed and the presentation of some results concerning the propagation of a solid-solid combustion front. In section two, we present the mathematical modelling. Section three is devoted to the description of the temporal discretisation method. Section four outlines the main features of the finite volume scheme applied to the enthalpy balance. Section five details
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Finite volumes for complex applications
the conservative, spatial adaption procedure. Section six compares the computational cost of the finite-volume schemes in one, two and three dimensional numerical simulations of the propagation of a stiff combustion front.
2. Mathematical Modelling
2.1 Chemical kinetics We denote by 1 (T — T a/ o) the Heaviside function, and consider the one-step exothermic kinetics whose rate follows an Arrhenius law for Ti + C —> TiC, with an ignition temperature Tna. The conversion rate Y behaves like:
2.2 Enthalpy balance and boundary conditions The total enthalpy of the system, takes into account two isothermal phase changes. The first one occurs at T=Tag = 1166K and the second one at T=TSl = 1943 K with the respective latent heat effusion Lap and Lsl. Denoting by TO, a fixed reference temperature, the enthalpy of the system exhibits a discontinuity at T = Tap and T = Tsi, [BON 73], and writes
The governing enthalpy balance equation can be cast in the form
In cartesian coordinates, we have the following boundary condition
The stiffness of the problem, both from spatial and temporal point of vue, requires an adaptive procedure to compute accurately and at a low cost the shape and the velocity of the front.
Adaptivity, tracking and fitting 827 3.0 Temporal scheme
The temporal discretization is based upon a Backward Difference Formula of order 2 with a non-uniform timestep, which handles efficiently such stiff systems. This is a three-level scheme , second order accurate finite-difference pmr>rriYirn atinn
An adaptive procedure based upon cite9 is implemented to dynamically increase or decrease the time-step. The reaction is only active after the first phase transition at 1166K, therefore during the first part of the process, below that critical temperature, the time step is around O . l s . In the second part of the process, when the reaction is active, it is decreased down to approximatively 0.001 s. We remark that the chemical kinetics equation can be written in a more eeneral form as
with U(T) = V (T) = £ 0 .exp (-&) -1 (T - T a/3 ). We notice that an > 0,/? n < 0,7 n > 0. We have the "explicit" equation of the conversion ratio at the new timestep, which is iteratively solved by a fixed point Picard method.
4.0 Spatial discretization scheme
We emphasize below a three-dimensional finite volume discretization principles. The reader is referred to [AOU 99B],(resp. [AOU 99C]), (resp.[AOU 99D]) for the detailed description of the scheme in one , (resp. two), (resp. three) dimensions and extensive discussion about numerical simulations. 4.1 Computational domain and numerical problem We describe in this section the numerical methodology used in the threedimensional case with the computational domain f2 — [0, L] x [O,L] x [0, H] plit i = I j = J k=K
into a finite number of parallelepipedic volumes such that £2 = IJ U U ^i,j,k, i =1 j = 1 k = 1 on which the enthalpy balance is integrated. We assume having cell-centered
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Finite volumes for complex applications
formulation, where a constant unknown is located at the center Cj-j^ of the parallelepipedic cell &i,j,k = [ # j ) # j + i ] x [yj,yj + i] x [z/c, z^+i], with lengths a,- = xi+i - Xi,bj - yj+i - yj,ck = zk+i - z fc ,and volume mtj = L . dft = > 31 ^ (ai.bj.Ck • The finite volume method is well suited for the discretization of such parabolic equations expressed in conservation form, with Neumann boundary conditions [GAL 92]. We must smooth the Heaviside function used to express the phase change, and introduce the phase fraction of phase a0, denoted by fa/3 £ [0)1]) and the phase fraction of phase si, denoted by /5/ £ [0,1]. We have 1
The discontinuity of the enthalpy at the phase transition temperatures Ta/3 and Tsl is replaced by a small intervall over which the phase change occurs. This is achieved by defining T%,p = Tap — e, and Tlaj5 = Tap -f e, where 6 = 0.01 x Tap is the phase change intervall. In fact, this method introduces a "mushy zone", which is ^-dependent. Assuming that the density is constant throughout the process, and omitting the details, we are led to the following formulation
v-v We are now able to formulate, the numerical problemby integrating the enthalpy balance over the time-space finite volume [tn,tn+i] x Ki j
An iterative update procedure for the liquid phase fractions fap and fsl is defined identically by:
It has been shown in [SHY 94], that an efficient stable update procedure can be devised by substituting the previous equations directly into the enthalpy balance to get an another relationship namely the T_based method update.
Adaptivity, tracking and fitting
829
4.3 Laplacian discretization We apply Green's Formula, an denoting Bn+1 = B (rn+1,Yn+1) ,, we obtain
We must now, discretize the integrals and evaluate the thermal conductivity at the interface between cell Qij,k and QJ+IJ^. We apply the mean formula, and evaluate the integrand at the enter of the cell fijj^.We compute the diffusion fluxes at the interface between the cells Qijtk and QJ+IJ^ by a centered approximation,first order(resp. second order ) accurate on a nonuniform (resp. uniform) mesh. This requires the computation of the distance between the centers of two adjacent cells. An important issue, is the computation of the thermal conductivity at the interface between the cells ^i,j,k and ^i+ij,A:- We use the continuity of the diffusion flux at the interface to write an harmonic mean relationship for the thermal conductivity at the interface. This ensures the conservativity of the diffusion flux across the interface. 4.4 Matrix formulation After some simple, but rather tedious algebraic calculations, we are led to matrix formulation
We get an heptadiagonal , nonsymmetric, strictly dominant matrix which is inverted by the simple straightforward Jacobi iterative method.
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Finite volumes for complex applications
5.0 Spatial adaption scheme In order to maintain the structure of the mesh, we consider an extremely simple, straightforward, easy to implement adaptive procedure based on a finite volume discretization of the reaction-diffusion. In multi-dimensional simulations, it is applied in the same way, successively to the x,y,z or r,z directions. It has the advantage, of using the same matrix solver throughout the numerical simulation. We describe here the main features of this method along the x-axis in the three-dimensional case. We assume given a parallelepipedic cell Qij^, and compute an adaption criteria to decide whether to refine or unrefine the finite-volume mesh. 5.1 Adaption criteria We assume given $m (value of the criteria for unrefinment)and <&M (value of the criteria for refinment). We compute the adaption criteria, based on the curvature of the temperature solution, along the x axis, where a and 0 are given positive weights, and decide wether to refine or coarse the mesh.
5.2 Refinment procedure If $jis greater than <$M, then we split £li,j,k U fii+ij^into three equal volumes, along the x-axis, compute the new value U of the field U ( for example T, Y ), at the second cell such that the field U is conserved by:
We have a linear, barycentric combination of t/;and C/i+i. The sum of the weights affected to t/iandt/i+iis equal to 1 and we have the following property.
5.3 Unrefinment procedure If 3>jis smaller than <£ m , then we fuse
fijj^Ufij+ij^into
one single volume,
Adaptivity, tracking and fitting
831
along the x-axis, and compute the new value U of the field U ( for example T, Y)by:
We have a linear convex combinaison and therefore £/,- G [min ([/,-, C/,-+i) , max (Ui, Ui+1
6.0 Numerical simulations and discussion All the numerical experiments presented in this paper were conducted on a Power Challenge and on the Origin 2K of Crihan. The various programs were coded in C programming language and were run in 64-bit precision. We present a table of the computational gain obtained by the use of adaptive methods. Id
N=100 nodes 10 s N=100 nodes, A< adapt 7s A£ adapt, Ax adapt
2d
N=100 nodes 1053 s N=100 nodes, A* adapt 289 s A£ adapt,, Ax adapt 240s N=100 nodes 10530 s N=100 nodes, A* adapt 2955 s At adapt,, Ax adapt 2500 s
N=200 nodes 18s N=100 nodes, Ai adapt 12 s
N=400 nodes 42s N=100 nodes, Ai adapt 26 s
N=200 nodes 3033 s N=200 nodes, A* adapt 1080 s
N=400 nodes 13779 s N=400 nodes, Ai adapt 6510 s
N=200 nodes 32000 s N=400 nodes, Ai adapt 9140 s
N=400 nodes 150 000 s N=400 nodes, Ai adapt 42 000 s
4s
3d
Figure 1. Temperature profile along X axis Figure 2. Temperature profile along X, Y- axis
7.0 Conclusion and perspectives A variety of Simula tion codes in (ID, 2D and 3D) has been written to investigate numerically the relationship between heat supply - surface of application and combustion front propagation[AOU 99E]. Work is in progress to determine the most efficient matrix solver in two and three dimensions. Acknowledgment Extensive numerical computations were performed on Origin 2K, through grant 199009 from CRIHAN.
832
Finite volumes for complex applications
[AOU 99A]
AOUFI A., VREL. D., PETITET J.P., "Modeling of SelfPropagating Synthesis of TiC", Intern. Journal of SelfPropagating High-Temperature Synthesis, 6, 1, 41-53, (1997).
[AOU 99B]
A.AOUFI, "Self-Adaptive Finite-Volume Scheme for reactiondiffusion equations with Phase Change. Application to the numerical simulation of one-dimensional self-propagating fronts", submitted to Journal of Computational and Applied Mathematics.
[AOU 99C]
A.AOUFI, "Fully Adaptive Finite-Volume Scheme for reaction-diffusion equations with Phase Change. Application to the numerical simulation of bidimensional self-propagating fronts", submitted to Int. J. Num. Meth. Eng.
[AOU 99D]
A. AOUFI, "Three Dimensional Fully Adaptive Finite-Volume Scheme for Reaction-Diffusion Equations with Phase Change. Application to the numerical simulation of self-propagating fronts", submitted to Applied Numerical Mathematics.
[AOU 99E]
A. AOUFI, D. VREL AND J-P PETITET. "Bidimensional Numerical Analysis of Critical Parameters for Stable Combustion Synthesis Study of TiC", Gordon Research Conference on Oscillations and Dynamic instabilities in Chemical Systems, June 6-11, 1999, I1 Ciocco, Barga, Italy. Accepted for Presentation (Poster).
[BON 73]
BONACINA C., COMINI G., FASANO A. AND M. PRIMICIERO /'Numerical SOlution of Phase-Change Problems", Int. J. Heat Mass Transfer., 16, 1825-1832, (1973).
[GAL 92]
T. GALLOUET, "Methodes des Volumes Finis", Collection Problemes Non Lmeaires Appliques, CEA-EDF-INRIA, 2830 Octobre 1992.
[SHY 94]
W. SHYY AND M.M. RAO, "Enthalpy based formulations for phase change problems with application to g-gitter", AIAA paper No. 93-2831.
[VER 94]
VERWER J.G. "Gauss-Seidel iteration for stiff odes from chemical kinetics", Siam J. Sci. Computing. Vol 15. No. 5, pp. 1243-1250, September 1994.
[VRE 95]
VREL D., LIHRMANN J.M., PETITET J.P., "Synthesis of titanium carbide by self-propagating powder reactions: part 1, enthalpy of formation of TiC", J. of Chem. and Eng. Data, 40, 280-282, 1995.
Mathematical and numerical modeling of a two-phase flow by a Level Set method. Sandra ROUY1 and Philippe HELLUY2 1 DCNINGENIERIE CENTRE SUD S/D LSM B.P. 30 83800 Toulon Naval FRANCE 2 ISITV - Laboratoire Modelisation Numerique et Couplages B.P. 56 83162 La Valette du Var CEDEX FRANCE
ABSTRACT This study is devoted to the numerical simulation of a two-phase flow. Because of its formal simplicity, a Level Set method has been chosen. The interface is thus located by the zero level set of a smooth function which is convected at the speed of the fluid. A conservative system can also be written which is solved by the Finite Volume method. Special care is needed for the pressure law at the interface between the two fluids. Some numerical results are presented. Key Words: two-phase flow, finite volume, level set, conservation laws, Roe's method
1
Introduction
The purpose of this study is to describe the separated and time-dependent gasliquid flow which appears in the cooling chamber of a gas generator where the temperature can reach 2500 K and the pressure 180 bar. The time scale is about one microsecond. We have two objectives. The first one is to follow the evolution of density, velocity, pressure and temperature in both fluids. The second one is to accurately capture the shape of the interface separating the two fluids. The aim of this research is also to implement a simple method in order to achieve these goals. The most common model for describing two-phase flows is the two-fluid flow model obtained via a space and/or a time average of the local instantaneous equations of conservation laws for each phase (ref [SAIN 93]). These equations have then to be coupled with some transfer terms between the phases. This model is especially concerned with dispersed two-fluid flows because the geometry of this type of flows allows a formulation of these transfer terms. In this way a system is obtained which can be easily solved.
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Finite volumes for complex applications
On the contrary, when dealing with separated two-phase flows, it is more judicious to use an interface tracking method. Two different approaches are possible: the lagrangian approach and the eulerian approach. In the lagrangian methods the interface is located by marker particles (ref [UNV 92]). These methods are accurate and close to physical phenomena. However they require a lot of particles when the interface is complex. Moreover regridding algorithms must be employed in order to properly track the interface. The eulerian methods are divided into two categories: the VOF methods and other related methods (ref [HIR 81]) and the LEVEL SET methods (ref [MUL 92]). The Volume-Of-Fluid methods introduce a new variable: the volume fraction of one of the fluids defined in each cell of the grid. This function is equal to 1 if the cell is filled with fluid and equal to 0 if it is empty. For a mixed cell which contains both fluids: the VOF function lies between 0 and 1. A specific numerical process is used to compute the transport of this quantity and to ensure the conservation of both liquid and gas phase mass. This process requires to reconstruct the interface at each time step. Different techniques of reconstruction, more or less complicated, have been proposed. These methods are especially used for incompressible fluids with free surfaces. In the Level Set methods the interface is represented as the zero level set of a smooth function (j> defined on the entire physical domain and satisfying a transport equation. This function is initialized as the signed distance from the free surface. In this formulation the interface can merge or break up with no special treatment. It presents several advantages: there is no need to reconstruct the interface or to regrid the neighbourhood of the discontinuity. Moreover it can be easily generalized to three dimensions. For these reasons, we have chosen a Level Set method. This type of method seems to be the simplest way to treat the interface. In all the methods described above, it is very difficult and sometimes impossible to deal with two-phase flow with a large density ratio. Our main concern is to overcome this difficulty.
2 2.1
Governing equations Equations of motion
We consider all phases as a single fluid which is supposed to be compressible and inviscid ; we neglect gravity, phase changes and surface tension. The fluid is then governed by Euler equations:
where p is the density of the fluid, it the velocity, E the specific total energy, P the pressure and e the internal energy with the relation E — e + |U2.
Adaptivity, tracking and fitting
2.2
835
The level set formulation
The interface F is then located by the zero level set of (j> (ref [MUL 92]). F is defined by F = {x/ (x, t] = 0}. The level set function is positive in the gas and negative in the liquid. Hence we have:
2.3
Closure of the system
The system is thus the following:
In order to close this conservative system, an equation of state is needed which must be relevant to the different states of the fluid. This is the difficult point of the modeling. We use the ideal gas equation of state for the gas and the Stiffened-gas equation for the liquid (ref [COC 96]). Therefore the pressure depends on the density, the internal energy and 0. The equation of state is then: P = P ( p , e , 0 ) = a ( 0 ) p e + B() P - (j - l) pe with 7 = 1.4 if 0 > 0 P = (N - 1) pe - TV Poo with N = 5.5 and P^ - 4.92115E8 Pa if 0 < 0 We can already notice when internal energy is small, the Stiffened-gas equation provides negative pressure. We shall see later that this is one of the major difficulties of our model.
3
Resolution of the system
We solve system (1) in two dimensions by the Finite Volume method which is well-adapted to the equations of conservation. A first order Godunov-type method which consists in solving Riemann problems between cells was chosen as a starting point for investigations (ref [GOD 96]).
836
Finite volumes for complex applications
At the interface between two cells, we shall solve a problem of type:
with velocity of the fluid,
is the and
Because of its simplicity we decided to use Roe's method. This method consists in a linearization of the above system that leads to solve (y is the outward normal to the interface):
Let us recall the main features of this method. A system of type (2) admits a Roe-type linearization, if for any (UL, UR) in E x E , there exists two matrices A1 (UL, UR) , AZ (UL,UR) , such that the three following points are verified:
(property of conservativity) the Jacobian matrix of f1 the Jacobian matrix of f2 (property of consistency) (iii) for any direction v = (v1,v2) 6 R such that \v\ — 1, the Roe matrix Av (UL,UR) "in the direction of i/" defined by v1A1 (U L ,U R )+v2A 2 (UL,UR) has real eigenvalues with a complete family of eigenvectors (property of hyperbolicit-y). In practice, the Roe matrix is determined by computing an averaged state between UL and UR. We apply a change of variables W ->• U(W) such that U(W) and / (U(W)) are homogeneous quadratic functions of W and so we can write the following relation:
In our case, we can prove the existence of the Roe matrix defined by (iii) where the average state is given by:
Adaptivity, tracking and fitting
837
Then A1 and A2 write:
A,(U)
A2(U)
The pressure law coefficient a is equal to 7 — 1 in gas and to N — 1 in liquid. The main difficulty here is to properly calculate a in such a way that it produces a correct pressure at the interface between the two fluids. Several choices will be discussed in section 4. The eigenvalues and the corresponding eigenvectors of Roe's matrix are given by:
that is to say in
c is the sound speed and is equal to
H is the total specific
gas and in liquid c enthalov which satisfies the relation H = E
Roe's scheme associated to the linearization is then:
where £]* represents the summation over all edges e of the control volume Ei, ne is the normal direction to the edge and Uf is the approximation of the average value of U (., t} on the control volume. The numerical flux is given by: A+ and A~ are defined by the relation A± = TA ± T min max and For stability, a CFL condition has to be respected:
l
where A1*1 = diag
max
838
Finite volumes for complex applications
4
Results
In order to validate Roe's solver, we compute the well-known test case of the shock tube which reduces to a one dimensional Riemann problem:
There are three characteristic families corresponding to the distinct eigenvalues. We can explicitly compute the one-parameter families of shocks, simple waves and contact discontinuities. The family of shocks is computed thanks to the Rankine-Hugoniot jump conditions which are the following:
The families of simple waves and contact discontinuities are computed thanks to the Riemann invariants which are u and P for the contact discontinuities and s (the entropy ), u ± ^c and 0 for the simple waves (ref [SMO 94]). We obtain via this resolution an interesting information: the level set function is equal to zero at the contact discontinuity. An example of the solution is shown on the following figure.
As a first validation, we consider a two gases flow where the pressure laws are different. Density, velocity and 0 are represented on the following figures.
Adaptivity, tracking and fitting
839
We can see that the capture of the contact discontinuity is not satisfying. Moreover, this leads to oscillations (ref [COC 96]). Another point is that we have to introduce an intermediate equation of state when
840
Finite volumes for complex applications
5
Conclusions
The level set method is a simple method to track the interface. Indeed we obtain a conservative system which can be easily solved by the Finite Volume method. Moreover, no special treatment is needed for tracking the interface. However in the case of a gas-liquid flow, we have to overcome the fact that we have two different types of equation of state. An intermediate law has to be introduced, which keeps physical sense. Moreover, the Stiffened gas equation has to be modified to avoid negative pressure. Finally, in order to properly capture the contact discontinuity, a second order scheme at least must be employed. Some results with a gas-liquid flow will be presented during the symposium.
6
References
[SAI 93] SAINSAULIEU, L.-An Euler System Modelling Vaporising Spray-Dynamics of Heterogeneous Combustion and Reacting Systems,1993,Vol.152,pp. 280-305. [UNV 92] UNVERDI, S.O. and TRYGGVASON, G.-A Front-Tracking Method for Viscous, Incompressible, Multi-fluid Flows-Journal of Computational Physics- 1992, Vol. 100,pp.! 37. [HIR 81] HIRT, C. W. and NICHOLLS, B.D.-VOF Method for the Dynamics of Free Boundaries-Journal of Computational physics,1981,Vol.39, Nfl,pp.201-225. [MUL 92] MULDER, W., OSHER, S. and SETHIAN, J.A. -Computing Interface Motion in Compressible Gas Dynamics-Journal of Computational Physics,1992, Vol.100,pp. 209-228. [COG 96] COCCHI, J.P., SAUREL, R. and LORAUD, J.C.-Treatment of Interface Problems with Godunov-Type Schemes-Shock waves,1996,Vol.5,pp. 347-357. [GOD 96] GODLEWSKI, E. and RAVIART, P.A. -Numerical Approximation of Hyperbolic Systems of Conservation laws-Applied Mathematical Sciences,1996- 509p. [SMO 94] SMOLLER, J. -Shock Waves and Reaction-Diffusion equations- SpringerVerlag,1994-632p.
Multiresolution analysis on triangles: tion to conservation laws
applica-
A. Cohen, S.M. Kaber, M. Postel* Laboratoire d'analyse numerique Universite Pierre et Marie Curie (Paris 6), France http://www. ann.jussieu.fr
ABSTRACT A multiresolution algorithm is coupled with a finite volume scheme to solve scalar bidimensional conservation laws. The originality resides in the adaptivity of the multiresolution decomposition, which takes into account the possible appearance of discontinuities and their displacement. Numerical simulations on triangular meshes point out the advantages of the method in terms of CPU and memory costs. Key Words: Multiresolution - finite volumes - adaptive scheme.
1. Introduction We are interested in this work in solving conservation laws on polygonal domains. It is well known that such equations can develop localized discontinuities in finite time. In such areas where the solution is not smooth a fine resolution is necessary and furthermore, higher order schemes for flux computations are necessarily nonlinear - including for instance ENO reconstruction. These are very costly techniques and it seems reasonable to use all available information on the local smoothness of the solution to decide whether they should be used or not. This can be done within multiresolution framework. The original idea of combining the advantages of this method - data compression, smoothness indicators - into conservation laws solvers in order to reduce the number of flux computations is due to Harten [Har94]. Initially implemented in one dimension, it was then extended to bidimensional cartesian grids, and smooth deformations of rectangular grids, (see references in [CDKP99]). Preliminary approach for unstructured grid is explained in [Abg97]. In the case of triangular meshes, which are more flexible in modeling complex geometries, a complete implementation with a detailed analysis of the encoding/decoding algorithm can be found in [KP99, CDKP99]. In this former approach, the solution is represented everywhere on a uniform fine grid and the multiscale analysis is used to speed up the flux computations in the smooth areas. What we propose here is a fully adaptive scheme which
842
Finite volumes for complex applications
make use of the compressed solution - encoded at each time step by its most significant coefficients. This approach gives way to new difficulties, for instance in the analysis of the stability and precision. These points have been studied in detail in the one dimension case in [CKMP99] and are currently under investigation in the triangular meshes case. The outline of the paper is as follows: we recall in section 2 the multiresolution and the finite volume algorithms for triangular meshes which are then combined to produce the adaptive scheme presented in section 3. Numerical simulations and analysis of performances on tests cases follows in section 4. 2. Multiresolution analysis and finite volumes
We briefly describe a multiscale transformation of a function described by its mean - and not point wise as usual - values on a triangular mesh. The equation is defined on a polygonal domain fi which is discretized by a coarse grid f2° made of 7V~o triangles. A hierarchy of nested grids $lf (0 < t < L) is built. The grid $V+1 is obtained by dividing each triangle of & into four smaller triangles by joining the three edges midpoints. Triangles from the level i are denoted by T% for 1 < k < Ng = 4*NQ. We denote by uek the average of a function u on the triangle T| and ul = (w()^ 1 . The mean values ui+l of a function u being given on the ftl+1 grid, the mean values on the coarser grid £le can be computed by
where T/+1 are the four triangles of £ll+l obtained by subdividing the triangle T/. The center triangle is denoted by T^J"1. The relation (1) defines a projection operator Pg+l from the resolution level i + 1 on the level t by P^+lue+l = ue. Multiscale decomposition relies on the existence of this operator as well as a prediction operator Q^+1 from the resolution level i to the level I + 1. They should be compatible in the sense that Pii+lQ(i+l = Id. The predicted values ui+l — Qll+lul are approximations of the exact mean values uf+l obtained by local linear combinations of values at level t. Details on every triangle are defined as the differences between exact and predicted values d\ j — u**1 — u^1, for j = 1,... 3. The multiscale representation of UL is denoted by UMR = (w°,d°,... ,dL~1}. Going from UL to UMR and reverse is achieved through wavelet-type algorithms described in [CDKP99]. The two types of data representation have the same memory requirement. However, in the multiscale representation case, the amount of data can be reduced by thresholding the sufficiently small details. More precisely this consists in setting dek • = 0 wherever \d£k j\ is smaller than a level dependent tolerance £t. We refer to [CDKP99] for justification. In the same reference, the stability and
Adaptivity, tracking and fitting
843
convergence of a class of reconstruction operators are studied. This leads to the following cfptimal choice for Qee+l
We now recall how the numerical treatment of the ID scalar conservation law
can be improved within the multiresolution framework. The unknown u(x, y, t) is defined on a polygonal set fi, for t > 0. An initial condition u(x,y,0) = uo(x,y} is imposed as well as boundary conditions on the edges of ft. We first consider that equation (3) is solved by a finite volumes scheme on the fine grid r& L . The costly step of this scheme is the flux evaluation T>k'n ~ 1 r I div f(7Z.)dxdy. Since the solution can present discontinuities - even L T \ k \ JT£ with smooth initial data - the computations may have to be locally very precise. On the other hand, the point wise evaluation of a function known by its mean values requires a reconstruction step (using the operator Tl in the flux evaluation, not to be mistaken with the prediction Q^+1 defined by (2)). This step can also be very costly if ENO techniques are used for instance. It is therefore crucial to limit the number of these flux computations and this is where multiscale analysis comes into play. In Marten's framework, the details provided by the multiscale decomposition of the solution are used as criteria to choose the flux computation method as follow. Starting from the coarsest level where fluxes on all triangle edges are precisely computed. The levels are explored in turn: wherever details are large, fluxes are computed using precise and costly numerical flux approximations Vlk — —j- y^ \ ^ i , j \ f k , j l-Tfcl j
wnere
'•— 5^ l^ml/m- w^ fm denoting the numerical flux across the r cr ~ Ledge r^, (of length |r^|) between two triangles belonging to the finest grid fJ L . Elsewhere, the mean values of flux divergences are computed by interpolating the same quantities on the immediately coarser level, using the prediction operator Q f + 1 . This algorithm was successfully implemented in [KP99, CDKP99]. It leads to a CPU reduction by a factor 2 to 3 depending on the test case. Its precision is governed by the precision of the underlying finite volume scheme on HL since at each time step the time evolution of the solution is performed on this grid. Therefore the gain in CPU time remains linked to this discretization, \^k,j\fk,j
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Finite volumes for complex applications
and there is no memory saving. A way to overcome these two limitations is provided by a fully adaptive scheme in which the solution is computed on a nonuniform time dependent grid. In the next section we modified the initial scheme in order to compute the solution on such a hybrid grid, fine where large fluctuations are foreseen and coarse in the smooth areas. We refer to [CKMP99] for a detailed analysis of the algorithm in the one dimensional case. We are principally interested here in showing the potential of the method in the case of multidimensional grids - and more precisely triangular ones. 3. Fully adaptive schemes
We introduce a hybrid grid *H™ C U^fl^ defined at time step tn by selecting subsets out of each f^. We will show how this grid is built in such a way that it reflects the smoothness of the solution. Since this regularity is time dependent, the hybrid grid will have to evolve in time, this will also be described. Eventually we will explain how the finite volume scheme is performed on such a grid, so that both the solution mean values and the details in the multiresolution analysis evolve correctly in time. We first describe how %° is built using the initial data UQ. It is initially set equal to fi°. Until the maximum level of resolution if reached, each triangle of H® is tested to decide whether its subdivisions should be added to 1{®. The criterion is the comparison of the mean value of the solution on the four subdivisions with the values predicted by interpolation Q^ +1 - A smoothness indicator Rek, associated to a triangle T^, is activated (Rfk — 1) if one of the four details dlk . is greater in absolute value than the threshold tolerance ££. The triangles whose indicators are activated are actually divided into four triangles who are themselves included into the computing grid H°. Another requirement is to preserve the grid nestedness. As a corollary we must always be able to interpolate at the finer level. This has two consequences: • If a triangle T^ is divided because Rlk = 1, all its neighbors on the same level i are included in 7^!, even if their smoothness indicators are not activated. • An auxiliary grid Q® containing "H° is defined: If a triangle belongs to the grid %j? for one of the two previous reasons, then all its neighbors at level t belong to Q®. On triangles belonging to G®\HQ£ no actual computation takes place and the mean values of the solution will be estimated by Q^+1. When the initial data is given by its dual multiresolution representation, pathological cases with non zero details only on the fine level can occur. To handle such cases, all the triangles should be actually divided up to the finest level, even if the details are negligible at a coarser level, and the thresholding should be done iteratively starting from the finest level. This is in practice very difficult to realize without giving up all the advantage of the multiresolution in terms of memory savings. In the test cases we have studied so far, the initial solution is given by an explicit expression, which ensures that the tree structure of H° is achieved by the previous algorithm which explores and refines the levels
Adaptivity, tracking and fitting 845 starting from the coarsest. The hybrid grid may have to be modified at each time step in order to follow the discontinuities propagation and capture shocks formation. To this effect, we define a larger grid %™ +1 , containing "H™, whose time evolution can be predicted. We use here the hyperbolicity of the problem ensuring that if the CFL condition is respected the discontinuities do not propagate further than one space step in one time step. To foresee possible propagation in any direction the term of neighbors of a triangle is extended to all triangles sharing a vertices with it, and not only the three triangles having a common edge and participating in the interpolation scheme Qlf+l. A corresponding grid £™ +1 containing "H™+1 as well as all its neighbors at each level is also defined. At each time step the mean values of the solution on the coarsest grid and the details corresponding to triangles in 14™ are modified (as detailed in algorithm 1). Whenever one of the four details of a triangle of 14™ is higher than the level tolerance, its smoothness indicator and that of all its neighbors is activated. If a detail is higher than a still larger tolerance, the corresponding indicators at the finer level are also activated. These operation can make triangles previously in Q™\H™ part of 7-Ln+l and even create new subdivisions at the finer level. The nestedness of the new grid i-Ln+l is ensured as in the initial step - which can induce the creation of subdivisions in Q™+1. Eventually some triangles of Qn\14™ may now be useless and therefore removed. An important difference with Marten's algorithm is that the computing grid H™ is incomplete. In other words, the depth of resolution is spatially variable and defined for a triangle T^ of level t by a local depth I. This has direct consequences on the precision of scheme. In both approaches the precise numerical flux evaluations are performed only on the interfaces of the hybrid grid. In Harten's case the solution is everywhere available on the finest grid so the maximum precision can be achieved. In the adaptive case, the solution could theoretically be computed down to the finer level by applying the Q^+1 as many times as necessary starting from I where it is actually known. This would be completely inefficient in practice since it would destroy all the benefits of the multiresolution. On the other hand, brutal use of the available mean values on the hybrid grid to compute the fluxes leads to an order of approximation governed by the coarser level. This problem is explained in details in the one dimension case in [CKMP99] and we simply summarize here the main conclusions. We have so far two ways to tackle this difficulty: • In one dimension it is possible to reconstruct directly - that is linearly mean values at the finest level from values at a not immediately coarser level as soon as the details are null on a sufficient number of consecutive intervals on this coarse level (four intervals in the case of linear reconstruction). Taking into account the areas of high gradient where actual decoding must be performed leads to a complexity in Nlog(N} where N is the cardinal of 14™. • The second method consists in using a higher order ENO type scheme to compute the fluxes at points where the solution is known by its mean values on
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Finite volumes for complex applications
a coarse or intermediate grid level. In that case the complexity is that of the hybrid grid multiplied by a fixed coefficient which depends on the complexity of the ENO reconstruction. The precision is that of the coarse grid to a power related to the ENO reconstruction order. From the parameter study performed in [CKMP99], it seems that the first method is the most efficient one if we compare the performances for a given accuracy. However its extension to the triangular grid case seems more complicated. In this preliminary implementation we have used the same ENO type reconstruction as in [CDKP99]. We summarize in the following adaptive algorithm (1), how the finite volumes scheme is used to compute the time evolution of the solution and the details of the multiresolution on the hybrid computing grid Une. Algorithm 1 Finite volumes + adaptative multiscale Initialization of 1~L® and u°MR Loop on time step n = 0,1,... Determination ofH™+1 Partial decoding Evolution on the coarse grid: with Loop on levels with
4. Numerical experiments
To illustrate the performances of the algorithm we first present the test case of a linear flux on an initial discontinuous solution. We choose the function Uo(x,y) = 1 if \fx2 + y2 < 0.1 and zero elsewhere on the unit square with periodic boundary conditions. Performances in term of CPU time, memory requirements and precision are evaluated for the adaptive multiresolution scheme and the reference finite volumes scheme on the finest level. The numerical fluxes are computed with the second order ENO method already implemented in [KP99, CDKP99]. It requires a small CFL number chosen here equal to 0.2 on the finest grid. There are 50 triangles on the coarsest level. We first show results corresponding to a multiscale decomposition on a maximum of 5 levels. In this case the finest grid on which the finite volumes reference solution is computed counts 12800 triangles. The direction of propagation is parallel to
Adaptivity, tracking and fitting
847
the Ox axis and the solution is computed over one period of time. Figure 1. shows the initial solution and the solution after one period computed with the adaptive scheme on a maximum of five levels with a total of 1209 triangles. The shape of the solution is well preserved, and the finer levels are used only in the vicinity of the discontinuity. On the next group of figures 2, the performances of the adaptive scheme are displayed for different maximum number of levels. For each value of this parameter L, the standard finite volume solution on the finest grid - level L - is computed. CPU time and memory requirements along with the error with the exact solution in the LI norm can be compared. The precision of the adaptive scheme in this case is very satisfying since the two error curves are always close to each other. Actually, in this test case, the source of possible errors is the discontinuity and in its vicinity both schemes are similarly discretized. As far as the memory requirement are concerned the advantages of the adaptive case are indisputable. We are actually restricted to a few number of levels because the finite volume scheme will not run on our work station for a grid finer than the 5th level (L = 4). We can predict that for 204800 triangles it would require about 13 hours of CPU. In this limit case L = 6, the adaptive scheme actually discretizes the solution on 2198 triangles, among which only 1228 belong to the finest grid and takes only 1.5 hour to run. One remarks that the CPU time does not depend linearly upon the number of triangles because some expensive overhead computing must be done which is inherent to the multiscale decomposition like the creation and the deletion of triangles. The adaptive scheme is nevertheless much faster than the standard finite volume scheme, all the more so as the number of levels in the multiresolution increases. 5. Conclusions A fully adaptive scheme based on the multiresolution decomposition of the solution is proposed to improve the performances of a standard finite volume algorithm. The mean values of the solution are computed on a time varying grid made of cells belonging to different levels of decomposition according to the local smoothness. The preliminary numerical tests which are presented exhibit very good performances in term of CPU and memory savings. The immediate future works consists in developing the vector case algorithm necessary to handle gas dynamics. From a more theoretical point of view, we are currently studying the convergence of the scheme. Bibliography [Abg97]
R. Abgrall. Multiresolution analysis on unstructured meshes: application to CFD. In Experimentation, modelling and combustion. John Wiley & Sons, 1997.
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Figure 1: Multiresolution solution on five levels at time t=0 and t=l (one period)
Figure 2: Comparison of CPU times, memory occupation and precision for finite volumes and multiresolution algorithms [CDKP99]
A. Cohen, N. Dyn, S.M. Kaber, and M. Postel. Multiresolution schemes on triangles for scalar conservation laws. Submitted 1999.
[CKMP99]
A. Cohen, S.M. Kaber, S. Miiller, and M. Postel. Adaptive multiresolution finite volume schemes. In preparation, 1999.
[Har94]
A. Harten. Adaptive multiresolution schemes for shock computations. J. Comp. Phys., 115:319-338, 1994.
[KP99]
S.M. Kaber and M. Postel. Finite volume schemes on triangles coupled with multiresolution analysis. CRAS 328, serie 1:817,822, 1999.
A local level set method for the treatment of discontinuities on unstructured grids L. Tran, R. Vilsmeier, D. Hanel
University of Duisburg, FB7 / IVG, Germany E-mail: {linhbao, hjOOOvi, hj454ha}@ivg.uni-duisburg.de ABSTRACT The paper is concerned with the treatment of discontinuous problems in field simulations on arbitrary grids. The location and movement of the discontinuities are described by a level-set function with restricted, dynamic definition range. The physical transport of the discontinuities is decoupled from normalizing the level set function within its definition range and an error correction mechanism is employed to minimize unphysical effects. The back influence from discontinuities to the continuous part of the field simulation is provided by flux separation, where no sub-cell resolution is required. Object oriented programming based on MOUSE allows an arbitrary number of discontinuities for arbitrary physical effects within the same computation. Key Words: discontinuities, fitting/tracking, local level set, 00-programming
1. Introduction
In the past decades remarkable progress was made concerning the efficiency and accuracy of capturing methods for the field simulation of problems with embedded discontinuities. Corresponding work was mainly motivated by computations of compressible flows where discontinuities together with continuous solutions are resolved by the same conservative method. In all capturing methods the discontinuities are not represented as sharp jumps but they are smeared over a certain number of nodes, which leads to artificial, intermediate states. In many problems, in particular where large scales have to be resolved, these effects may be acceptable. However, if strong interaction between discontinuities and the surrounding field are present, these artificial states can remarkably falsify the solution. Typical examples for that are reacting flows, like detonations or flames. Front tracking enables numerically a representation of discontinuities as jumps and tracks them in space and time. Although the mathematical and physical principles of weak solutions of conservation laws are well known, the numerical realization is complex and has lead to different approaches. Among the different ideas for tracking a front the level-set approach [KER 88, MUL 92, SUS 94] was found to be well suited to develop a tracking concept on arbitrary grids.
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2. Governing equations and solution concepts
Consider the following general system of equations modelling a physical problem of interest: |-
/ QdV+ i H-ndA = I S dV (1) V
A
V
where Q, H and S represent the variables, fluxes and source terms respectively. The system of equations is solved on arbitrary grids employing a nodal finite volume method, figure 1. A discrete formulation of the above system reads as follows:
Figure 1: Control volume
where Vd is a control volume with the size Vvd and the sum is carried out over all bounding segments. As underlying software, the MOUSE package [GLO 97], currently in development at the cite of the authors, is used. The following methods for front-tracking are, however, not yet available in the present public release. 3. Method for front propagation
An embedded discontinuity is a feature, that is one dimension below the actual problem considered. It is a point in a 1-D problem and a line or a surface in a 2-D or 3-D problem respectively. An obvious trial is to transport the discontinuity by a Lagrangian approach through the mesh. This approach works well in 1-D, but yields some problems in 2-D and 3-D, since the topology of the line or surface must be respected, which is not evident. To overcome this drawback, an Eulerian formulation can be introduced, yielding the well known level-set methods, (]. Before proceeding, we may classify the discontinuities in two distinct types: Passive discontinuities, which are simply transported by a given carrier speed. Referring to fluid dynamics, this is the speed of the fluid and examples are material interfaces, contact discontinuities or free surfaces. Active discontinuities, which are able to move relative to a carrier. Examples in fluid dynamics are shocks or combustion fronts (if described discontinuously). Comparing both types, the remarkable difference for a computational approach is, that the propagation speed of active discontinuities is only defined on the discontinuity, while the carrier speed governing the motion of a passive discontinuity is available everywhere in the field. 3.1 Description of the location and motion of discontinuities
In the present work a level-set formulation is used on a fixed computational grid. The method introduces a continuous function G(x, [y, z ] , t ) , while the actual position of the discontinuity is given by a discrete iso-value, typically 0.
Adaptivity, tracking and fitting 851 The motion of the discontinuity can thus be described by a scalar convection equation:
where c is the propagation speed of the discontinuity. Figure 2 sketches such a discontinuity on a grid, consisting of arbitrary elements. For this and the following subchapters some sets S are defined here. Points are referred to by the letter P, edges by K and levels by L. For example, a set of edges around a node i is referred to as SK-P(i), a Figure 2: Discontinuity on a 2-D set of nodes in level / by SP-L(l) and the set mesh and membership of nodes and of nodes neighbouring a node j by SP-P(j). edges to corresponding levels, Pm e Furtheron the level l of a node i or an edge k SP.L(m) and Km € 5A'_L(m) is defined as LP(i) or LK(k] respectively. For the present paper, discrete values for G are stored at the nodes of the mesh. It is however not required to define the function G in the whole computational domain, instead it can be described locally in two neighbourship levels only. The definition of a level for an edge k G SK-L(m) reads:
and the edges hitting the front are of level 0. For a node i G SP-L(m) we write:
The restriction to two neighbourship levels imposes, that for each step of motion, only nodes i G SP-L(l) can be overrun, if any. In this case, the function G can still describe the position between the first and the second level. After each time step, the definition range is adjusted accordingly. The above equation (3) is discretized in space and integrated in time. For a space discretization employing a finite volume method, propagation speeds are needed everywhere on the surface of the control volumes. Unfortunately, these are not available in the case of active discontinuities, thus requiring a corresponding extrapolation already at the nodes i G SP-L(l). Due to this drawback, a differencing variant is preferred. The term c- VG can be directly evaluated for all edges k G SA'_L(0). The gradient VG can easily be computed on all nodes i G SP-L(l) employing a Gauss-type integration on the finite volumes or a least squares method as shown below. A simple directional upwinding is applied in the direction of the edges, combined with a central formulation for the transverse component due to lack of information. Since the discrete values for G are stored at the nodes of the mesh a redistribution of the time derivative 4^, computed at the edges k G S7i_L(0) onto the appending nodes has to be performed. As the propagation speed is generally not available at edges k £ S/\_L(0), a discretization on these is per-
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Finite volumes for complex applications
formed with extrapolated speeds. This residual is however only required, if the corresponding node in the first neighbourship gets overrun. 3.2 Normalization of the level set function The precision of the method is directly related to the curvature of the Gfunction normal to the discontinuities. Therefore it is useful to provide a constant slope as far as possible. This is the aim of normalization. It is desired, that the function G satisfies the normalizing condition |VG| = C, where C is a constant ^ 0, typically chosen C = 1. This equation corresponds to the description of a sand hill, where the position of the discontinuity can be thought to be the outer cut line at constant altitude. To achieve normalization, several methods can be used. Modifying the basic equation (3) by adding a normalizing contribution controlled by a switching function S, with S = 0 for G = C, yields:
This formulation has shown to be less useful here. Although the original equation is restored for 5" = 0, the physical transport is falsified, since the nodes and edges for which S is applied are not located right at the discontinuities. Upon the experiences with the present work it is therefore preferable to separate the physical transport and the normalization completely. Assume that equation (3) has been solved for a time step. A normalization of the G-function can now be achieved with a pointwise iteration process for the nodes in the definition range around the discontinuity. The iteration process starts at the nodes of the first neighbourship level SPJL(l), iterating up to convergence, and then proceeds from level to level in increasing order. This method corresponds to parabolic space marching although the problem is in fact hyperbolic. Within a level, the pointwise iteration is carried out by a two stage method, based on least squares. In a first step, the direction of the gradient VGn is computed, in the second step the functional value of Gn+1 is set to satisfy the slope condition. Step 1: Consider the following least squares ansatz:
with rhij = Xj — Xi and a distance weighting exponent q > 0. The formulation is one-sided due to the restricted consideration of neighbours in Epar(i). Differentiation and minimization yields the following system (exemplary for 2-D):
allowing the computation of a one-sided gradient VG? at iteration level n.
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Step 2: Again with least squares, but now projected in the direction of the gradient VGJn, the following ansatz is obtained:
where Considering d^hyp/dG\n
' — 0 for minimization and setting C = I yields:
For the nodes i G SP-L(l) this approach must be modified, since the position G = 0 on the edges k G (SK-P(i)r\SK-L(Q)) crossing the discontinuity must be preserved. For these edges, the actual neighbour node j on the opposite side of the discontinuity is replaced by a phantom node j* at the crossing position with G(j*} = 0, see figure aside. Since the distance |m;j.| can approach 0 as the discontinuity moves across the mesh, it is useful to expand the fraction with the minimal distance |mTO,n|9 found, thus replacing the terms 1^-77 by \?ln\\ , and these terms hold in the limit \mmin\q -> 0. Also ^ j € E h y p ( i ) 'p^.jf > 1, since the addend fc^|I is contained. Additional error correction: The above described normalization can still alter the position of a discontinuity. Since depending on the curvature, a shift of the front line appears, an additional correction method is required. For this, the exact position of the front before normalizing the nodes i G SP-L(l) is stored and compared to the position after the normalizing step. Considering a linear representation of G over the edge, an error correction is done, computing an increment AG/c for the edges, that satisfies:
where A/ error is the shift after the iteration in the first level is completed. Since several edges k G 5/\_L(0) may be appended to a node i G SP-L(l) the corresponding correction is taken as average. Although this additional correction is simple, its effect is very remarkable as shown in figure 3. 3.3 Propagation speed, back influence and overrunning nodes
In the previous chapter a propagation speed cfwas used, without stating how this is obtained. Considering a passive transport, the carrier speed is easy to
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Finite volumes for complex applications
Figure 3: Passive convection of a circular line on a field with constant carrier speed. Comparison with and without additional error correction. Computed solutions.
access. For active discontinuities, the propagation speed depends on the corresponding values at both sides of the discontinuity. Consider as example a shock, whose propagation speed can be computed upon the pressures at both sides. For the edges k G S7\ _L(0) the left and right states can be computed upon projected variables from the ending nodes. Since the position and motion of discontinuities can be treated as described, it is now interesting, how the discontinuity affects the surrounding fields. In the present work, a flux separation scheme was used, not requirering a sub-cell resolution. In this method the original control volumes of the mesh are preserved. Referring back to figure 1, each edge of the mesh carries a segment of two adjacent control volumes. In smooth regions, a projection from the nodes storing the variables to the cell interface is performed and a flux computed in a central or upwind manner. The flux is Figure 4: Principle of the fluxthen used for both the adjacent control volumes. separation method. For all edges k 6 SK-L(Q) this is no longer the case. The flux for the "left" side is computed using the "left" side projection, and the the flux for the "right" side with the right side projection accordingly, see figure 4. This method is very simple and can be employed on any grid. As a node of the mesh gets overrun by a front, the variables stored at the corresponding nodes switch to the other side. At present these new values are obtained by extrapolation from the the surrounding neighbours at the new side. Note, that this method does not ensure conservation. However, a fully conservative variant of the method is currently in development and expected to be available soon. 4. Computational examples The above presented methods have been developped without any preference for a specific application. However, the examples below are related to fluid dynamics. The figures 5 and 6 show computed results for a shock tube problem where both, the front shock and the shear layer are tracked. The computation
Adaptivity, tracking and fitting
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was performed on a 2-D unstructured mesh with triangular elements. As a second example, the inviscid, transient computation to steady state for a cylinder at Ma^ — 2 is shown. As initial condition a straight shock is located in front of the cylinder. The computation is performed on the grid shown in figure 7. Figures 8, 9 and 10 show the tracked position of the bow shock, the corresponding subset grids and isolines of density for the initial condition, an intermediate and the converged solution.
Figure 5: Shock-tube example: Isolines of density, subset meshes and location of shock and contact discontinuities. Initial condition (top), after 300 time steps (below) and zoom of the contact discontinuity after 300 time steps (left). These figures are computed results.
Figure 6: Shock-tube example: cut through a 2-D solution at different time levels.
References [KER 88] A. KERSTEIN, W. ASHURST, F. WILLIAMS: Field Equation for Interface Propagating in an Unsteady Homogeneous Flow Field. Phys. Rev. A, vol. 37, pp2728-2731, (1988). [MUL 92] MULDER, OSHER, SETHIAN: Computing Interface Motion in Compressible Gas Dynamics. J. Cornp. Phys. , vol. 100, pp 209-228, (1992). [SUS 94] M. SUSSMAN, P. SMEREKA, S. OSHER: A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow. J. of Comp. Physics, vol. 114, pp 146-159, (1994). [GLO 97] O.GLOTH, R.VILSMEIER, D.HANEL: Object oriented programming for computational fluid dynamics, HiPer' 97, Krakow, Poland, (1997).
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Figure 7: Top: computational mesh Figure 8: Series at top, right: Tracked position of the bow shock, initial (left), intermediate (middle), and final state (right). Figure 9: Series at right: Subset meshes, initial (left), intermediate (middle), and final state (right).
Figure 10: Series at right: Density isolines, initial (left), intermediate(middle), and final state (right).
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A Stabilized Version of the Wang's Partitioning Algorithm for Banded Linear Systems * Velisar Pavlov Center of Applied Mathematics and Informatics, University of Rousse, 7017 Rousse, Bulgaria
Abstract The parallel partitioning algorithm of Wang for arbitrary nonsingular banded systems is stabilized. Some numerical experiments (including random matrices) are presented. Key words and phrases. Iterative refinement, parallel partitioning method, banded systems, perturbation.
AMS(MOS) subject classification. 65G05, 65F05, 65Y05
1
Introduction
Banded systems of linear equations appear in many problems and are the computing time consuming kernels of various applications. The systems arise either directly, as in the difference approximations of ordinary differential equations, or after suitable rearrangement of equations and unknowns, as in finite element methods for elliptic problems. Such systems we can solve in parallel by the so called partition methods. A typical member of these methods in the case of tridiagonal systems is the method of Wang [8]. This method gives an efficient parallel algorithm for solving such systems. Full roundoff error analysis for the whole algorithm in the case of nonsingular tridiagonal matrices is presented in [10]. Generalized versions of the partitioning algorithm of Wang for banded linear systems are presented in [2, 6]. Full roundoff error analysis in this case can be found in [11]. In this work it is shown that the algorithm is numericaly stable for some special classes of matrices, i.e. diagonally dominant (row or column), "This work was supported by Grants 1-702/97 and MM-707/97 from the National Scientific Research Fund of the Bulgarian Ministry of Education and Science.
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Finite volumes for complex applications
symmetric positive definite, and M-matrices. Unfortunately when the matrix (even though well conditioned) of the system does not belong to the above mention classes, the algorithm can breack down or behave poorly. In our paper we present a stabilized version of the generalized Wang's algorithm for arbitrary nonsingular banded linear systems. Let the linear system under consideration be denoted by
where A € 7£ nxn , which bandwith is 2j + 1. For simplicity we assume that the number of superdiagonals j is equal to the number of subdiagonals. The partitioning algorithm for solving (1) can break down when it is necessary to divide by numbers which are less than a certain limit 6. In such cases we improve the algorithm perturbing the inputs or intermediate data. But the result which we get is perturbed. In order to make the solution more accurate we use iterative refinement (see [3]). Hence, it is necessary to solve (1) several times with different right hand sides. A similar perturbation approach is used in [1] for a Strassen-type matrix inversion algorithm, and in [4] for a fast Toeplitz solver. The convergence of the iterative refinement is analysed in [7, 9]. The outline of the paper is as follows. Section 2 presents the partitioning algorithm. In the next section we consider perturbations and iterative improvement of the solution. Finally, in Section 4 we present some numerical experiments (including random matrices) in MATLAB.
2
The partitioning algorithm
For simplicity we assume also that n = ks — j for some integer k, if s is the number of the parallel processors we want to use. Let us note that our assumptions are not essential for the consideration. We partition matrix A and the right hand side d of the system (1) as follows:
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where B{ 6 72. (fc J) x ( f c J\i = 1,2, . . . , s , are band matrices with the same bandwith as matrix A, a»,Ct are matrices of the following kind
whose elements a( i _ 1)fe+1 ,Ci fc _i 6 7# XJ ', aik,bik,Cik G njxj ,i = 1, 2 , . . . , s-1, and finally
Now we define the following permutation
of the numbers [1,..., sk—j], and denote the corresponding permutation matrix by P. By applying this permutation to the rows and columns of matrix A we obtain the system
wherein = diag{Bi, 52, • • • ,B,} e n'(k-3)*>(k-3\
here A2l e T^-1)**^), A22 = diag(6 fc ,6 2fe ,..., 6 (s _ 1)fe ), e The algorithm can be presented as follows. Stage 1. Obtain the block LU-factorization
by the following steps:
ft*-1)**-1).
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1. Obtain the LU-factorization of AH = PiL\U\ (with partial pivoting, if necessary). Here PI is a permutation matrix, LI is unit lower triangular, and C/i is an upper triangular matrix with diagonal elements u\ ,i4 , • • •, ua,'k .), using Gaussian elimination (with pivoting, if necessary) . 2. Solve using the LU-factorization from the previous item, and compute S = AH — A<2iR, which is the Schur complement of A\\ in A. Now let us notice that when we solve (2) it is necessary to divide by u\ for i ~ 1 , . . . , s(k — j). In this case if the blocks Bi (one or more) are singular, then at least one of the quantities u\ ' becomes very small or zero and the algorithm can break down, or behave poorly. In order to avoid this dangerous situation we propose to perturb them with 5, where S is sufficiently small. The implementation of this idea is presented in the next section. Stage 2. Solve Ly = d by using the LU-factorization of AH (Stage 1). Stage 3. Solve Ux = y by applying Gaussian elimination (with pivoting, if necessary) to the block S.
3
The Stabilized Algorithm
As was noticed in the previous section the algorithm can break down, or behave poorly, when wt- , for i = 1,..., s(k — j) and are zero or small. So, we can perturb them in such a way that it would be away from zero. The stabilization step can be summarized as follows:
In this way we shift u\ away from zero. Hence, the algorithm ensures that we do not divide by a small number.
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Prom the other side the obtained solution is perturbed. Then we apply the usual iterative refinement from [3], with some modification:
The difference here is that instead of A we solve perturbed systems with the matrix A+A, where A is a diagonal matrix with all such perturbations, and x is the result of the perturbed algorithm before the iterative refinement is applied. We note that, when 8 = ^/p~Q « 10~8 (in double precision), in practice the perturbed solution is very close to the exact one and we need usually only one or two steps of iterative refinement, depending on what accuracy we require. Here by PQ we denote the machine roundoff unit. Taking into account [7] the condition of convergence of iterative refinement is Ccond(A)6 < 1, where cond(A) is a condition number of matrix A and C is a constant of the following kind maxidAHxDi l o —
4
; 77-771 i\~i mm»(|A||z|)i
~ -!•> ^? • • • ? ' * •
Numerical Experiments
Numerical experiments in this section are done in MATLAB, where the roundoff unit is PQ K 2.22 x 10~16. The exact solution in our examples is x = (1,1,..., 1)T, by NH we denote number of iterations, and we measure two types of errors: 1. The relative forward error
where x is the computed solution; 2. The componentwise backward error (see [5])
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Table 1: The forward and backward error and the number of iterations of Example 1, when e — 0
We stop the iterations when at least one of the following conditions is fulfilled:
2. Number of iterations is > 10; Let us consider the following examples. Example 1. Let A be a matrix of the following kind:
This matrix is very well conditioned. In our tests we set E equal to 0 and 10 12. Hence, the blocks Bi become singular. We report the results with different values of 6 in Table 1 (when E = 0) and in Table 2 (when e = 10~12). When S = 0 we obtain the original algorithm without stabilization. We see that the original algorithm can break down (when e — 0) on account of dividing by zero, or behave poorly (when E = 10~12). At the same time the stabilized algorithm gives much better results. Let us note that at most two steps of iterative refinement are enough and FE, BE are sufficiently small when 6 = 10~8, and that only a few perturbations are necessary. Example 2. In this example we generate 1000 random matrices for each version of the algorithm (original and stabilized), where k — 7, s = 15, j = 2.
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0.42 1.15 0
6 = 10-7 1.16 x 10~13 6.54 x 10~12 1
6 = 10~8 1.33 x 1CT14 4.42 x 10~14 1
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6 = 10-9 6.44 x 1CT14 7.02 x 10~13 1
Table 2: The forward and backward error and the number of iterations of Example 1, when e = 1CT12
6 =0 Orig. Stab.
Average FE Worst FE Average FE Worst FE Average Nit Worst NH
6 = 10-7
6 = 10~8
6 = 10~9
2.42 x 10~12 3.11 x 10~9 1.41 2
2.22 x 10~13 2.52 x 10~12 1.12 2
4.23 x 10~13 6.72 x 10~n 1.24 2
0.12 2.77
Table 3: The forward error and the number of iterations for the random matrices of Example 2.
Then the blocks Bi are matrices of the following kind:
Now let us fix where i is a random integer which belongs to the range [1,15]) to make the block B{ singular and hence the original algorithm numericaly unstable. We report the results in Table 3 for different values of <5, again. The second and the fourth rows of Table 3 present the average forward error for the two versions of the algorithm, and the third and the fifth rows contain the worst forward error which is obtained during the series of experiments. The average and the worst number of iterations we present in the sixth and the seventh rows of Table 3. The tests with random matrices show that the optimal value of 6 is 10~8, again. So, if we compare both algorithms, we see that the stabilized algorithm is significantly better than the original one. In conclusion, the algorithm behaves well in pracrice, and we need to do more research for its theoretical justification.
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References [1] Balle, S., Hansen, P., Higham. N.: A Strassen-type matrix inversion algorithm for the connection machine. APPARC PaA2 Deliverable, Esprit BRA Contract # 6634; Report UNIC-93-11, UNI«C, October 1993. [2] Conroy, J.: Parallel Algorithms for the solution of narrow banded systems. Appl. Numer. Math. 5 (1989) 409-421. [3] Golub, G., Loan, C. van: Matrix Computations. The John Hopkins University Press, 1996. [4] Hansen, P., Yalamov, P.: Stabilization by perturbation of a 4n2 Toeplitz solver, SIMAX, 1999. (to appear) [5] Higham, N.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia, 1996. [6] Meier, U.: A parallel partition method for solving banded linear systems. Parallel Comput. 2 (1985) 33-43. [7] Skeel, R.: Iterative refinement implies numerical stability for Gaussian elimination. Math. Comp. 35 (1980) 817-832. [8] Wang, H.: A parallel method for tridiagonal linear systems. ACM Transactions on Mathematical Software 7 (1981) 170-183. [9] Yalamov, P.: Convergence of the iterative refinement procedure applied to stabilization of a fast Toeplitz solver. Proc. Second IMACS Symposium on Iterative Methods in Linear Algebra, Eds. P. Vassilevski and S. Margenov, IMACS (1996) 354-363. [10] Yalamov, P., Pavlov, V.: On the Stabilty of a Partitioning Algorithm for Tridiagonal Systems. SIAM J. Matrix Anal. Appl. 20 (1999) 159-181. [11] Yalamov, P., Pavlov, V.: Stability of a partitioning algorithm for special classes of banded linear systems. Preprint N 38, University of Rousse, March 1998. (submitted to LAA)
On Jeffreys Model of heat conduction
by Maksymilian Dryja and Krzysztof Moszyiiski University of Warsaw Banacha 2, 02-097 Warszawa Poland
ABSTRACT We present differential equations describing the Jeffreys Model of heat conduction, and certain corresponding numerical models. We define some suitable weak form of Jeffreys equations with boundary conditions of low regularity, (defined by L? functions), and formulate existence and uniqueness theorem for this generalized differential problem. We give also theorem on unconditional stability of the numerical model and propose its finite-volume interpretation. Some results of numerical computations, compared with experiments with heat waves are presented.
1.
Introduction
Classical Fourier law relates a heat flow Q and temperature T in a very simple way: where k is a positive constant. However, it is well known that the classical heat conduction equation based on this law does not describe well enough all phenomenons observed in experiments. Such phenomenon is, for example, so called heat wave incitated in thin metallic layer by a short heat impulse. Jeffreys model is based on a somewhat more complicated relation than that given by the Fourier law: it is certain integro-differential relation which involves so called relaxation time ([1],[2], [3],[4]). Due to the modified Fourier law, Jeffreys model fits better to experiments with heat waves: numerical computations based on this model show quite good consistency with experimental results. The heat impulses incitating the heat waves, which we observe in numerical experiments, are represented in the numerical model by the boundary conditions for Jeffreys differential equations. We apply very sharp, and even discontinous heat impulses. Differential equations of the Jeffreys model are in fact very
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simple, and the only difficulty in present problem was caused by the assumed low regularity of the boundary conditions.
2. Jeffreys Model and its weak form
General equations of Jeffreys Model can be presented in the following form:
where the scalar function T(t, x) is the temperature, the vector-function Q(t, x) is the heat flow, t £ R+, x G f2, £1 C R3, D and AC are positive coefficients (in general constant). In fact we are interested in one-dimensional model, because it corresponds to known results of experiments with the heat waves. The one-dimensional equations are of the following form:
with t e [0, tmax], x e [L, P], T(0, x) = T ° ( x ] , £(0, x) = Q°(x), T(t, L) =
and
and let f(t, x) = T(t, x) - <j>(t, x) and Q(t, x} = Q(t, x) - if>(t, x). Functions f and Q satisfy nonhomogeneous Jeffreys equations and homogenous boundary conditions. Further, we introduce new functions
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and
Using these two functions we have the following boundary value problem:
with zero initial and boundary conditions, where / and g are in terms of 0, ^>, T° and Q°. In order to state the weak formulation for equations (2), we are proceeding in quite standard way. We introduce two bilinear forms:
where:
Let:
We define now the space H of pairs of functions (R,S), in which we want to find solutions of our problem. Let H be the space of all pairs of functions
such that:
(R,S),
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Weak formulation of Jeffreys Model We are looking for a pair (R, S) € H such that for almost all t £ [0, tmax], for all V £ L2(L, P] and W G H%(L, P):
holds. The following theorem establishes an unique solution of the problem (3). Theorem 1. Assume that:
then in H there exists a solution (R,S) of the variational equation (3). Let U C H be the subspace o/H of such pairs (R,S) that
If a solution (R,S) o/(3) is in U, then it is unique in U. Proof of this theorem will appear in [5]. Now, let us give some comments on the above result. First of all it implies that the solution (R, S) of equation (3) is such that: and
exactly what we need for our solution (T,Q). Moreover, the function T with respect to x, is in L2(L,P), hence it should only weakly depend on the function <j> defining the boundary condition for it. This fact was observed during experimental computations.
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3. Numerical model In fact the used numerical model is an implicit finite difference scheme on the rectangular grid of points (2 n ,£fc), where t n = rn for n — Q , 1 , - - - , N with tmax = NT, xk = L + kh for k = 0,1, • • •, M + 1, and P = L + (M + l)/i. On this grid is defined the following finite difference scheme:
with the following initial and boundary conditions:
where TjJ1, Q£ are values of the grid functions in the point ( t n , X k ) , A/& = fk+i ~ fk, V/fc = fk - fk-i, A = £ and p = p-. The next step is to prove the stability of the scheme (4). In order to avoid strong regularity hypotheses concerning functions 4> and ip such as, for instance, boundedness of divided differences and so on, we proceede in similar way as in the section 2. We begin with functions f% = T£ - <j>(tn,xk), and Q£ = Q% - ^(in,xk], where
Using that, the scheme is defined as:
where:
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and
The grid functions R% and S% are subject to homogeneous Dirichlet boundary conditions, and the following initial conditions:
and Let and
For the scheme (5), the following stability theorem can be proved:
Theorem 2 If [i = p- is arbitrary, but constant, independent of r and h, then for any m, 0 < m < N — I, solution (Rn,Sn) of the problem (5) satisfies the inequality:
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where K depend only on
Theorem 2 implies in particular that if // is arbitrary, but constant, then T£ and Q^ are bounded by K in the sense of discrete L2 norm. This means unconditional stability and, by Lax Theory, convergence follows.
4. Numerical experiments There have been done a lot of numerical experiments involving equations (4). Further we compared graphs obtained by computation with those obtained by physical experiments. We succeeded to choose good enough values for coefficients D and K. We made computations with Fourier model as well. Conclusions are as follows: 1. Jeffreys model fits quite well to experimental results. The heat waves appear in both cases: when the impulse is introduced by initial or by boundary conditions. 2. If we see some kind of waves in Fourier model (it is possible only when the impulse is introduced by boundary condition), they travel with visibly too great speed, and their amplitude vanishes very quickly. 3. Coefficient D plays the most important role from the point of view of the speed of the heat waves. For more informations see [4]
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References [1] Brorson S.D. Fujimoto J.G. Ippen E.P. "Femtosecond electronic heattransport dynamics in thin gold films" Phys. Rev. Lett. 59 (1987),1962-1965. [2] Joseph D.D. Preziosi L. "Heat waves" Rev. Mod. Phys. 61 (1989),41-73 [3] Joseph D.D. Preziosi L. Addendum to the paper "Heat waves" Rev. Mod. Phys. 62 (1990),375-391 [4] Moszyriski K. Palczewski A. " Assymptotic analysis of heat propagation models" submitted to Archives of mechanics [5] Dryja M. Moszyriski K. "On Jeffreys model of heat conduction" Prepared for publication
Investigation some method of cavitating jets
S.A. Ocheretyany and V.V. Prokof ev Department of unsteady hydrodynamics, Institute of mechanics Moscow State University, 119899, Michurinski pr. 1, Moscow, Russia
ABSTRACT The problem of multiphase liquid flow after ventilating cavity with internal pressure exceeding the ambient pressure and with low concentrations of gas (or vapor) bubbles in liquid is considered. The two-dimensional flow considered corresponds in one case to Chaplygin-Kolscher scheme, and in another one-to the problem of two symmetric jets collision with a cavity formation. The velocity profile at large distances behind the cavity is determined. The most intensive disturbances in the gas-liquid mixture flow are introduced by the gas injection into liquid in the neighbourhood of the tail point of the cavity. And the injection of small gas bubbles is most efficient there. The vapor injection into a cavity is not efficient for increasing the liquid flow momentum, but it can be used for generation of a cavitating jet. The account of multi-velocity effects in the two-phase medium equations is essential for bubbles of supercritical size, when the viscous friction effect is small. Key Words: cavity, bubbles, heat and mass transfer, two-phase flow, multivelocity effects
1. Introduction We shall model such a flow in a plane-parallel statement, considering, for example, the flow over a body (Figure la), the rear critical point of which is substituted by a finite stagnation zone (cavity) with smooth closure of jets. Such a cavity can be generated by means of artificial injection (ventilation) of vapor or gas into the cavity. Since the boundary of such a cavity is Taylor instable, the flow will be accompanied by intensive outflow of gas (or vapor) in form of bubbles and by the formation of a two-phase tail. The first example corresponds to flow over a convex cylinder according to
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the Chaplygin-Kolscher scheme [GUR 65] with formation of finite stagnation zone behind the cylinder with pressure exceeding the pressure in a free stream. Here one can use the well-known solution for a circular cylinder [GUR 65]; we, however, shall use a simpler solution for a flow over a plate (Figure la) [PRO 98]. Though such a scheme can not be applied for description of flow over a plate (with singularities at points E and F), it is quite suitable for studying the flow in the cavity neighbourhood. The second example presents the collision of two symmetric jets with formation of the stagnation region [GUR 65]. We shall suppose for simplicity, that the jet-leading channels are stretched up to infinity. In this case, while moving upwards along the stream, the flow becomes non-univalent, but we shall ignore this fact, since we consider the flow in the neighbourhood of jets' closure point B (Figure Ib) and in the outgoing jet. If the flow is reversed, then we arrive at the problem of "splitting" of the jet flowing onto the region with heightened pressure, which was considered in [YAK 87]. Near the cavity the flow is highly inhomogeneous, and the bubbles fall into the region of high pressure gradients; as a result, the bubbles motion with respect to the liquid becomes essential. When the multi-velocity effects are taken into account, the model is more complicated: the medium becomes anisotropic [VOI 75].
2. The main relations and results.
2.1 Two-Phase liquid flow with low concentration of bubbles formulations. To model for a rarefied bubble mixture [GAR 73], [VOI 75], which takes into account the two-velocity effects:
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Here p = p/(l — a) is the mixture density under the neglection of vapor (gas) density in a bubble (p^const is the liquid density), V(u,i>) is the massaverage mixture velocity, which is equal to the mean liquid velocity under the neglection of bubbles mass, Vi(wi,vi) is the mean velocity of bubbles in an elementary volume, a = 4/3?rna3 is the volume concentration of bubbles (n is the numerical density of bubbles), a is the radius of bubbles, which are supposed to be spherical. For a steady motion, supposing the slipping-related stress tensor Tij to be a tensor function of the slipping vector ^W(WX, Wy) = Vi-V one can write [VOI 75]
Further, FI is the hydrodynamic force acting on a bubble of variable radius moving with variable velocity [VOI 73], [YAK 73]; F2 is the viscous drag force. In the first example, where the bubble size is small and the relative velocity is low (the Reynolds numbers of relative motion Re w 1), the k = 4 value was accepted for the viscous drag force F2 (the Adamar-Rybchinskii formula). For the second example, where larger bubbles are considered and the Reynolds numbers of relative motion reach the values of the order of 103, the asymptotic Levich formula was applied [LEV 62] (k = 12). Further, /zj is the liquid viscosity coefficient (the viscosity is taken into account in the bubbles/liquid interaction process only), pk is the gas pressure in a cavity, v — 1 - for the isothermal process and v — 7 (7 is the Poisson adiabatic index) - for the adiabatic expansion-compression of gas, a is the bubble boundary velocity, £ is the surface tension factor. We shall suppose that the bubbles only weakly disturb the liquid flow. We denote: u — UQ + u',v = V0 + v',p — po + p', where u',v' -C ^/UJ+V^,p'
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where /(/) is determined from the boundary conditions. For the function of liquid phase's flow disturbance P = u'Uo + v'V0 + p1 /pi along the stream line of undisturbed liquid flow ^ = const one can write (transferring in equations [2] from variables (x,y) to variables (x,t/>) [OCH 98]):
In equation [5] functions Fx and Fy are determined for the known potential field of flow UQ , VQ and from the velocities of motion of bubbles U\, V\ in this field of flow. Further, after determining U\, V\ and a equation [5] is solved by finding a quadrature along the stream line ip — const. We shall suppose that bubbles have the same initial radius ao, the initial pressure is pg = pk and that the velocities of liquid and bubbles at the cavity boundary (0 < / < Lk) are equal. Then the initial conditions for the system of ordinary differential equations, governing the trajectories of bubbles in the flowfield of the liquid UQ, VQ take the form at t — 0:
where Xk, yk are the cavity boundary coordinates. Under such initial conditions the Jacobian of transformation of Eulerian coordinates to Lagrangian ones J zeroes at the initial time moment, since the cavity boundary is an envelope of trajectories of bubbles. By this reason at the cavity boundary the numerical concentration n (or a) can not be specified by a finite quantity. Nevertheless, the arbitrary function /(/) can be determined [OCH 98]. Indeed, for x -> oo the undisturbed flow tends to uniform one, the velocities of phases become equal. Hence, for x ->• oo V\ -» 0, U\ —> VQQ and J —> Voody/dl. Then we shall have from [4]
where qn is the specific numerical flux of bubbles through the cavity boundary. This relation uses the fact that the flux of bubbles in a stream tube of bubbles' medium is conserved. Thus, instead of specifying the distribution n(l) we shall determine function /(/) by specifying the distribution qn(l). The integral quantities are conveniently used for normalizing the distribution. These quantities are:
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where TV is the number of bubbles carried away per time unit from the whole cavity, Qgk is the integral volume gas flow rate from a cavity. Further, the numerical density and volume concentration will be referred to characteristic quantities: nc — VN L, ac = y^, where Qoo is the volume gas flow rate normalized with respect to pressure p^, L is the characteristic linear dimension of the problem. So, by solving a simple sequence of problems-the equation of dynamics of individual bubbles, the equation for concentration [4] and the equation for disturbances [5], we find in the linear approximation the field of the disturbance function P. In order to determine the disturbances themselves, one must determine the vorticity from function P, then introduce the stream function of undisturbed flow and then solve the complicated boundary-value problem for the Poisson equation taking into account the free boundary deformation in the linear approximation. However, if we make use of the conditions for x —> oc (in motion along the escaping jet) UQ —> VQO, VQ —> 0, p' —) 0, then P —» FOOW', i.e. in the linear approximation, along with the distribution of concentration of bubbles, we find the profile of a longitudinal disturbed velocity far behind the cavity. This profile can be used for calculating the integral characteristics of a flow-the force acting on a body flown over and the jet momentum variation. If, however, the velocity of bubbles motion relative to the fluid is low and the terms of the order of square of relative velocity may be neglected, then it follows from [5], that the disturbance P is determined by the distribution of a volume concentration of bubbles and by undisturbed pressure variation along the liquid stream line. If, however, the bubbles are so small, that they are "frozen" into a liquid and their dynamics may be neglected, then a becomes a function of a local pressure and stream function. For example, for the isothermal law of bubbles expansion a = ao^OPoo/Poj where function ao(^) characterizes the volume concentration distribution at some injection cross-section or in a free stream. Then we obtain from [5]:
where poiC0) ls the undisturbed pressure distribution at the injection crosssection (it is supposed that in a free stream P = 0). 2.2 The Chaplygin-Kolscher flow disturbance by bubbles escaping the cavity. Within the framework of the model, which takes into account the motion of bubbles relative to a liquid, we shall consider a symmetric flow over a plate with formation of stagnation region according to the Chaplygin-Kolseher scheme, rather than a rear critical point [PRO 98]. The exterior of a unit circle in the right half-plane is chosen as a parametric region (the correspondence of points is shown in Figure la). The field of velocities is specified in a complex form by
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formulae:
Here u is a complex parameter, V/t is the velocity at the cavity boundary, quantity k\ is related with the cavitation number a by expressions
All velocities are referred to the free stream velocity, the linear dimensions-to the width L of plate AB. In virtue of smallness of gas bubbles as compared to the characteristic dimension of the problem, we assume the pressure in a bubble to be equal to the ambient liquid pressure (the bubble pulsations are neglected) and the relative motion velocity to be low (the term of the order of aW2 is absent in the equations for disturbances). The isothermal law of bubble expansion is accepted. Possessing information on the disturbance of a longitudinal component of liquid velocity in a wake far behind a plate, we may determine the force acting on a plate in connection with gas rejection from a cavity. The calculations will be performed for the following values of parameters: Re = poLVoc/pi = 104, Eu - 2p00/p0V^ = I and 3, a = -0,25 and -0,585; the size of bubbles was chosen to be close to supercritical one (therefore, the relative velocities are low): ao/L = 0,01 and 0,08. For bubble size lower than the critical value the viscous drag forces are dominating. Two versions of bubbles' flux distribution over the cavity boundary were used: 1) the specific numerical flux of bubbles qn is proportional to x^ (qn = 0 at the cavity head and maximum-at the tail point Xk = Lk], and 2) qn is proportional to yk (qn is maximum at the cavity head and zero-at the tail point). 2.3 The disturbance of flow in the outgoing jet at symmetric collision of jets. The example considered above indicates that the greatest increase of liquid flow in a wake behind a cavity is achieved, when the gas bubbles are injected in the neighborhood of the tail point of a cavity. If the cavity has considerable size
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(cr -> 0) and the flow disturbance mechanism is mainly related with the flow in the cavity tail region, then the wake behind the cavity can be investigated with using the model flow shown in Figure Ib as an undisturbed flow. This flow can be considered as an asymptotics for a —>• 0 for the jet air screen [GUR 65], which is suitable for description of a flow in the neighborhood of jets' connection point (point B in Figure Ib) and in the outgoing jet. The flow can be described in the oarametric form bv formulae
Here w is a complex potential, TV is a scale constant. The parametric variable u = £,+irj varies in the half-band 0 < £ < Tr/2; 77 > 0. Quantities in [11] denote: ^o = Vk/Voo < 1, Vk is the velocity at the cavity boundary, VQQ is the velocity at the outer boundary of a jet. The coordinate origin z — x + iy — 0 corresponds to point u = 0. The cavity and jet surfaces are, respectively, £ = 0 and £ = Tr/2 r\ > 0 (with moving upwards along the stream the shape of stream lines in the physical plane tends to circles, and the flow becomes non-univalent). The part of the real axis x > 0 corresponds to 77 = 0; 0 < £ < Tr/2. We denote the jet half-width for x —>• +00 by A (then the scale constant N = 2A/7r). Unlike the previous problem, the bubbles may reach here the outer boundary of a jet, and we shall suppose them to intersect the boundary without obstacle. In calculating the disturbance in a wake behind the cavity we shall use the model of [l]-[3] with regard to the terms quadratic in the relative velocity. We specify the flow parameters, which are typical for the problem of artificial generation of cavitating jets [OCH 98]: the head pressure is p = 5 10,1 MPa, the pressure in a cavity is pk = 2 MPa, the ambient space pressure is pa = 0,1 MPa, A = 2 mm. We shall suppose the gas bubbles to be injected uniformly along the cavity surface (£ = 0,17 > 0), and the gas in bubbles expands adiabatically. Unlike the first problem, some part of bubbles intersect the outer boundary of a jet here, and, hence, we shall use for normalization the mean volume concentration of bubbles in the outgoing jet ac for x —> +00. At the upper asymptotic section (77 —> oo) the stream lines represent the circles, and the pressure on these lines is constant [OCH 98]. Besides, in the uniform injection of bubbles all two-phase medium's characteristics (the volume concentration, the velocity components in the coordinate system fixed with undisturbed stream lines of liquid) are constant along these stream lines. It can be shown that, under neglection of viscous terms, the right-hand side in equation [5] zeroes in this case, and, hence, the disturbance P for 77 2>> 1 may be ignored. Due to this reason, in specific numerical examples we have considered both the case of finite injection region (0 < / < L*) and of infinite one (/ > 0). In the latter case the upper boundary of calculated region (the maximum value lmax has been selected numerically (based on the absence of Imax influence on calculation results).
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2.4 Results and descussion. Figure 2. presents the profiles of longitudinal velocity far from a cavity (in calculations this is the cross-section x/L = 2) for various injection conditions. The zero u value for y = 0 is a result of scattering effect of the field of flow in the cavity's tail point neighborhood on the bubbles medium (the pressure gradient becomes infinite here). The fact is, that in motion along the zero stream line the disturbance value is P = 0 at the cavity surface, since pressure Po is constant here, and at the section behind the tail point (CD) P — 0, since the concentration of bubbles a is zeroing here. The latter fact takes place, if the distribution of injection at the cavity boundary [7] has no singularity at l->0. Figure 3 shows the profiles of a disturbed longitudinal velocity u at the crosssection x/A = 5 with the initial size of bubbles OQ = 0 , 3 mm (ao/A = 0,15) for the head pressure of 5 and 10,1 MPa (a = —0,387 and -0,19, respectively) and with infinite length of injection region. The dashed line shows the profiles associated with liquid's density disturbance only. The comparison indicates that the account of terms of the order of aW2 decreases the flow velocity and, hence, these terms play a part of dissipative ones, their contribution being increased as the jet flow velocity lowers. (This is due to increase of bubbles slipping velocity in the region of high pressure gradients). Figure 4 shows similar profiles at p = 5 MPa (
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3. Bibliography
[GUR 65]
M.I.GUREVICH, Theory of jets in Ideal Fluids, Academic Press, New York 1965.
[YAK 87]
Yu.L.YAKlMOV, "Limiting water flows", Mechanics and Scientific and Technical Progress. V.2 [in Russian], Nauka, Moscow 1987, p. 7-25.
[VOI 75]
VOINOV O.V., PETROV A.G. On the Equations of Motion of Liquid with Bubbles, [in Russian], Izv. Akad. Nauk SSSR, Prikl. Matem. i Mekh., Vol.39, N° 5, 1975.
[GAR 73]
GARIPOV R.M. Closed Equations of Bubbly-Liquid Motion, [in Russian], Prikl. Matem. i Teor. Fiz.(1973), N° 6.
[OCH 98]
OCHERETYANY S.A., PROKOF'EV V.V. Multivelocity Effects in High-Pressure-Gradient Flow of Dilute Bubbly Media, Fluid Dynamics, Vol.33, N° 1, 1998, p. 71-81.
[YAK 73]
YAKIMOV Yu. L. Forces Acting on a Small Body in Arbitrary Flux of Incompressible Fluid and Equations of Motion of a Two-Phase Medium, [in Russian], Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza, N° 3, 1973.
[VOI 73]
VOINOV O.V., PETROV A.G. Lagrangian Function of a Gas Bubble in a Non-Homogeneous Flux, [in Russian], Doklady Akad. Nauk SSSR, Vol.212, N° 5, 1973.
[LEV 62]
LEVICH V.G. Phisicochemical hydrodynamics, Prentice-Hall, Englewood Cliffs (NY), 1962.
[SED 87]
SEDOV L.I. Metody podobiya i razmerenosti v mekhanike, [in Russian], Moscow, Nauka, 1987.
[OCH 95]
OCHERETYANY S.A., PROKOF'EV V.V. Cavitation Initiation by Hot Vapor Injection into a Cold Liquid Jet, Fluid Dynamics, Vol.30, N° 1, 1995, p. 717-724.
[PRO 98]
OCHERETYANY S.A., PROKOF'EV V.V. Rare Multiphase Liquid Flow after Ventilating Cavity, [in Russian], Summury edition anniversary academician L.I.Sedov, Institute of Mechanics Publishing, Moscow, 1998.
SUMMURY 1. It is shown that in the linear approximation in volume concentration the velocity profile far behind a cavity can be determined without
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solving the full problem on liquid flow disturbance and without determining the disturbances of free boundaries. 2. The solutions of specific problems have shown that the most intensive disturbance of liquid flow by bubbles at fixed injected gas flow rate and cavitation number takes place, when the gas injection region is shifted toward the tail point of a cavity, and the injection of small bubbles is most effective. 3. The vapor injection into a cavity is not efficient for increasing the liquid momentum. The vapor is worth to be injected into the heightened pressure region with the purpose of jet saturation with cavitation bubbles, i.e. generation of a cavitating jet. 4- The account of multi-velocity effects in the two-phase medium equations is essential for bubbles of supercritical size, when the viscous friction effect is small. The terms of the order of a square of relative velocity play a part of dissipative terms in the case of gas bubbles, and the accounting of these terms considerable decreases the momentum of moderate-size bubbles effect on a jet.
Figure 1. Schemes of How in a physical plane (z = x + iy) and in the plane of parametric variable (u = £ + ir)) for the Chaplygin-Kolscher flow (a) and for a smooth closure of jets (b)
Varia
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Figure 2. Curves 1 to 3: a = -0, 2; 4La — -0, 585; I and 2 are the first and second versions of injection aQ/L = 0,1; 3-ao/L — 0,08
Figure 3. Velocity disturbance profiles for x = 5, a0 = 0,3 mm
Figure 4. Velocity disturbance profiles for x = 5, po = 5 MPa
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Figure 5. Volume concentration profiles for vapor bubbles
Figure 6. Profiles of function P for vapor bubbles
Index des auteurs
Abgrall R. 3 Abouziarov M. 247 AchdouY. 231 AfifM. 387 Agaphonov B.N. 743 Alcrudo F. 543 Al Moatassime H. 595 Amaziane B. 387 Angermann L. 223 AngotP. 215 Aoufi A. 437, 825 Azarenok B.N. 795 Bailey C. 507, 585 Baines M. J. 787 Balakrishnan N. 331 Ballmann J. 671 Benazzouz S. 663 Berkvens P.J.F. 499 Berthon C. 307 Berti G. 655 Bidegaray B. 483 Blanc P. 117 Bleecke H. 753 Bojarevics V. 453 Bontoux P. 559 Botchev M.A. 499 BouzoufB.525 Bovolin V. 475 Brenner G. 753
Brigadnov LA. 197 Brufau P. 255 BuBmann A. 607 Caltagirone J.P. 367 CaraeniD. 315 Castro-Diaz M. 809 Catalano L. 323 Chapin V.G. 663 Chassaing P. 663 Chen M. 803 Clerc S. 395 Cohen A. 841 Con way S. 315 CoquelF. 11,307 Cortes J. 445 CoudiereY. 125 Croft N. 585 Croisille J.-P. 239 Cross M. 453,507,585 Darwish M. 339 Deconinck H. 27 DednerA.491 Degrez G. 27 Dervieux A. 631,817 Dick E. 709 Drikakis D. 725 Dryja M. 867 DuboisF. 133
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EdelvikF. 141 El Ganaoui M. 559 Elmahi I. 525 Ewert R. 733 EymardR. 149 Fadlun E.A. 693 Feistauer M. 647 Felcman J. 647 Forestier A. 717 Fort J. 701 Fournier L. 623 FuchsL. 315 Fiirst J. 701 GallouetT. 149, 155, 189,215 Garcfa-Navarro P. 255, 809 George P.L. 817 GhidagliaJ.-M. 69,483 Glinsky-Olivier N. 411 Gloth O. 623 Goryachev V.D. 743 Grabs T. 297 Greenshields C.J. 467 Greiner G. 753 Grim G. 205 Haaland S.E. 683 Hakim A. 595 HalamaJ. 701 Hanel D. 849 Helluy P. 833 HerardJ.M. 717 HerbinR. 149, 155, 189,215 Hittinger J. 357 HolstadA. 163 Hubbard M.E. 787 Hughes M. 453 Ivanenko S.A. 795 Ivankovic A. 459, 467 JaphetC. 231 Jolly B.A. 615
KaberS.M. 841 Kaltenegger K. 533 Khoshyaran M.M. 551 Klein R. 41 Kolyvanov V.G. 743 Kozel K. 639, 701 Kumbaro A. 445 Lau Man Wai J. 403 Lazarov R.D. 51 Leary S.J. 787 Lebacque J.P. 551 Leboucher L. 453 LenkeL.J. 419 Leonardi S. 693 Leservoisier D. 631, 817 Lie I. 163 Lioen W.M. 499 Lotstedt P. 769 Louda P. 639 Lu H. 585 Lukacova-Medvidova M. 289
Maday Y. 231 Majewski J. 725 Maneeratana K. 459 Marmignon C. 11 Mazhorova O.S. 559 Meese E.A. 683 Meinke M. 733 Meister A. 297 Merci B. 709 Michel A. 173 Moczala M. 607 Morton B. 347 Morton K.W. 289 Moszynski K. 867 Moukalled F. 339 Miiller L. 533 NatafF. 231 NkongaB.575 Noelle S. 95 Ocheretyany S. 875
Index des auteurs OhlbergerM. 761 Oldroyd A. 567 Orlandi P. 693 Ouazar D. 525 Pasciak J.E. 51 Pavlov V. 859 PenanhoatO. 817 Pericleous K. 453, 585 Perot B. 263 Perpeet S. 429 Peters N. 803 PostelM.841 PotapovS.V. 271 PraveenC. 331 Prihoda J. 639 Prokofiev V.V. 875
Rad M. 279 Raghay S. 595 Ris V.V. 743 Rister Th. 733 RizkM. 615 Roe P. 279, 347, 357 RohdeC.491 Rokicki J. 725 Rosenband V. 437 Rouy S. 833 RumpfM. 205 Scanlon T.J. 567 SchallE. 411,631 Scheffler H.P. 181 Schmid O. 607 Schneider Th. 41 Shurina E. P. 377 Simon H. 419 Smirnov E.M. 743 Scares Frazao S. 403 Soderberg S. 769 Sonar T. 297
Stechbarth J. 533 Steelant J. 709 Taglialatela L. 475 Taylor G.A. 507, 585 Thiele F. 777 Toumi I. 445 Tran L. 849 Tremel U. 753 van Keuk J. 671 VanselowR. 181 Vassilevski P.S. 51 VerwerJ.G.499 Verzicco R. 693 VignalM. H. 189 VilledieuP. 125 Vilsmeier R. 849 Vincent S. 367 ViozatC. 631 VlassevaE. 517 Voitovich T.V. 377 von Lavante E. 429, 607 Warnecke G. 289 Weller H.G. 467 Wenke P. 567 WesenbergM. 491 Wheel M.A. 567 Wheeler D. 585 Wick A. 777 XeuxetE. 717
Ye X. 533 Zachcial A. 429 Zaitsev O.K. 743 Zech Y. 403 Zhang X. 263 Zoltak J. 725
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DEPOT LEGAL : SEPTEMBRE 1999. N° D'IMPRIMEUR : 46586. Imprime en France
